BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Jacob Tsimerman (University of Toronto)
DTSTART:20200408T190000Z
DTEND:20200408T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/1
DESCRIPTION:Title: Bounding torsion in class groups and families of local systems\nby Jacob Tsimerman (University of Toronto) as part of Harvard number th
eory seminar\n\n\nAbstract\n(joint w/ Arul Shankar) We discuss a new metho
d to bound 5-torsion in class groups of quadratic fields using the refined
BSD conjecture for elliptic curves. The most natural “trivial” bound
on the n-torsion is to bound it by the size of the entire class group\, fo
r which one has a global class number formula. We explain how to make sens
e of the n-torsion of a class group intrinsically as a selmer group of a G
alois module. We may then similarly bound its size by the Tate-Shafarevich
group of an appropriate elliptic curve\, which we can bound using the BSD
conjecture. This fits into a general paradigm where one bounds selmer gro
ups of finite Galois modules by embedding into global objects\, and using
class number formulas. If time permits\, we explain how the function field
picture yields unconditional results and suggests further generalizations
.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Kriz (MIT)
DTSTART:20200415T190000Z
DTEND:20200415T201500Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/2
DESCRIPTION:Title: Converse theorems for supersingular CM elliptic curves\nby Da
niel Kriz (MIT) as part of Harvard number theory seminar\n\nAbstract: TBA\
n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (BU)
DTSTART:20200422T190000Z
DTEND:20200422T201500Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/3
DESCRIPTION:Title: Modularity for self-products of elliptic curves over function fie
lds\nby Jared Weinstein (BU) as part of Harvard number theory seminar\
n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Newton (Kings College London)
DTSTART:20200506T190000Z
DTEND:20200506T201500Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/4
DESCRIPTION:Title: Symmetric power functoriality for modular forms\nby James New
ton (Kings College London) as part of Harvard number theory seminar\n\nAbs
tract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arthur-Cesar Le Bras (CNRS/Paris-13)
DTSTART:20200513T190000Z
DTEND:20200513T201500Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/5
DESCRIPTION:Title: Prismatic Dieudonne theory\nby Arthur-Cesar Le Bras (CNRS/Par
is-13) as part of Harvard number theory seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preston Wake (Michigan State)
DTSTART:20200520T190000Z
DTEND:20200520T201500Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/6
DESCRIPTION:Title: Tame derivatives and the Eisenstein ideal\nby Preston Wake (M
ichigan State) as part of Harvard number theory seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Stanford University)
DTSTART:20201104T200000Z
DTEND:20201104T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/7
DESCRIPTION:Title: A geometric approach to the Cohen-Lenstra heuristics\nby Aaro
n Landesman (Stanford University) as part of Harvard number theory seminar
\n\n\nAbstract\nFor any positive integer $n$\,\nwe explain why the total n
umber of order $n$ elements\nin class groups of quadratic fields of discri
minant\nhaving absolute value at most $X$ is $O_n(X^{5/4})$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karol Koziol (University of Michigan)
DTSTART:20201028T190000Z
DTEND:20201028T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/8
DESCRIPTION:Title: Supersingular representations of $p$-adic reductive groups\nb
y Karol Koziol (University of Michigan) as part of Harvard number theory s
eminar\n\n\nAbstract\nThe local Langlands conjectures predict that (packet
s of) irreducible complex representations of $p$-adic reductive groups (su
ch as $\\mathrm{GL}_n(\\mathbb{Q}_p)$\, $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)
$\, etc.) should be parametrized by certain representations of the Weil-De
ligne group. A special role in this hypothetical correspondence is held
by the supercuspidal representations\, which generically are expected to c
orrespond to irreducible objects on the Galois side\, and which serve as b
uilding blocks for all irreducible representations. Motivated by recent
advances in the mod-$p$ local Langlands program (i.e.\, with mod-$p$ coeff
icients instead of complex coefficients)\, I will give an overview of what
is known about supersingular representations of $p$-adic reductive groups
\, which are the "mod-$p$ coefficients" analogs of supercuspidal represent
ations. This is joint work with Florian Herzig and Marie-France Vigneras
.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arul Shankar (University of Toronto)
DTSTART:20201202T200000Z
DTEND:20201202T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/9
DESCRIPTION:Title: The 2-torsion subgroups of the class groups in families of cubic
fields\nby Arul Shankar (University of Toronto) as part of Harvard num
ber theory seminar\n\n\nAbstract\nThe Cohen--Lenstra--Martinet conjectures
have been verified in\nonly two cases. Davenport--Heilbronn compute the a
verage size of the\n3-torsion subgroups in the class group of quadratic fi
elds and Bhargava\ncomputes the average size of the 2-torsion subgroups in
the class groups of\ncubic fields. The values computed in the above two r
esults are remarkably\nstable. In particular\, work of Bhargava--Varma sho
ws that they do not\nchange if one instead averages over the family of qua
dratic or cubic fields\nsatisfying any finite set of splitting conditions.
\n\nHowever for certain "thin" families of cubic fields\, namely\, familie
s of\nmonogenic and n-monogenic cubic fields\, the story is very different
. In\nthis talk\, we will determine the average size of the 2-torsion subg
roups of\nthe class groups of fields in these thin families. Surprisingly\
, these\nvalues differ from the Cohen--Lenstra--Martinet predictions! We w
ill also\nprovide an explanation for this difference in terms of the Tamag
awa numbers\nof naturally arising reductive groups. This is joint work wit
h Manjul\nBhargava and Jon Hanke.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Marc Couveignes (University of Bordeaux)
DTSTART:20201209T200000Z
DTEND:20201209T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/10
DESCRIPTION:Title: Hermite interpolation and counting number fields\nby Jean-Ma
rc Couveignes (University of Bordeaux) as part of Harvard number theory se
minar\n\n\nAbstract\nThere are several ways to specify a number\nfield. On
e can provide the minimal polynomial\nof a primitive element\, the multipl
ication\ntable of a $\\bf Q$-basis\, the traces of a large enough\nfamily
of elements\, etc.\nFrom any way of specifying a number field\none can h
ope to deduce a bound on the number\n$N_n(H)$ of number\nfields of given
degree $n$ and discriminant bounded by $H$.\nExperimental data\nsuggest
that the number\nof isomorphism classes of number fields of degree $n$\nan
d discriminant bounded by $H$ is equivalent to $c(n)H$\nwhen $n\\geqslant
2$ is fixed and $H$ tends to infinity.\nSuch an estimate has been proved f
or $n=3$\nby Davenport and Heilbronn and for $n=4$\, $5$ by\n Bhargava.
For an arbitrary $n$ Schmidt proved\na bound of the form $c(n)H^{(n+2)/4}
$\nusing Minkowski's theorem.\nEllenberg et Venkatesh have proved that the
exponent of\n$H$ in $N_n(H)$ is less than sub-exponential in $\\log (n)$.
\nI will explain how Hermite interpolation (a theorem\nof Alexander and Hi
rschowitz) and geometry of numbers\ncombine to produce short models for nu
mber fields\nand sharper bounds for $N_n(H)$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Cass (Harvard University)
DTSTART:20200909T190000Z
DTEND:20200909T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/11
DESCRIPTION:Title: A mod p geometric Satake isomorphism\nby Robert Cass (Harvar
d University) as part of Harvard number theory seminar\n\n\nAbstract\nWe a
pply methods from geometric representation theory toward the mod p\nLangla
nds program.\nMore specifically\, we explain a mod p version of the geomet
ric Satake\nisomorphism\, which gives a sheaf-theoretic description of the
spherical mod\np Hecke algebra. In our setup the mod p Satake category is
not controlled\nby the dual group but rather a certain affine monoid sche
me. Along the way\nwe will discuss some new results about the F-singularit
ies of affine\nSchubert varieties. Time permitting\, we will explain how t
o geometrically\nconstruct central elements in the Iwahori mod p Hecke alg
ebra by adapting a\nmethod due to Gaitsgory.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao (CNRS/Harvard)
DTSTART:20201111T200000Z
DTEND:20201111T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/12
DESCRIPTION:Title: Frobenius and the Hodge numbers of the generic fiber\nby Zij
ian Yao (CNRS/Harvard) as part of Harvard number theory seminar\n\n\nAbstr
act\nFor a smooth proper (formal) scheme $\\mathfrak{X}$ defined over a va
luation\nring of mixed characteristic\, the crystalline cohomology H of it
s special\nfiber has the structure of an F-crystal\, to which one can atta
ch a Newton\npolygon and a Hodge polygon that describe the ''slopes of the
Frobenius\naction on H''. The shape of these polygons are constrained by
the geometry\nof $\\mathfrak{X}$ -- in particular by the Hodge numbers of
both the special\nfiber and the generic fiber of $\\mathfrak{X}$. One inst
ance of such\nconstraints is given by a beautiful conjecture of Katz (now
a theorem of\nMazur\, Ogus\, Nygaard etc.)\, another constraint comes from
the notion of\n"weakly admissible" Galois representations.\n\nIn this tal
k\, I will discuss some results regarding the shape of the\nFrobenius acti
on on the F-crystal H and the Hodge numbers of the generic\nfiber of $\\ma
thfrak{X}$\, along with generalizations in several directions.\nIn partic
ular\, we give a new proof of the fact that the Newton polygon of\nthe spe
cial fiber of $\\mathfrak{X}$ lies on or above the Hodge polygon of\nits g
eneric fiber\, without appealing to Galois representations. A new\ningredi
ent that appears is (a generalized version of) the Nygaard\nfiltration of
the prismatic/Ainf cohomology\, developed by Bhatt\, Morrow and\nScholze.\
n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Mantovan (Caltech)
DTSTART:20201021T190000Z
DTEND:20201021T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/13
DESCRIPTION:Title: p-adic differential operators on automorphic forms\, and mod p G
alois representations\nby Elena Mantovan (Caltech) as part of Harvard
number theory seminar\n\n\nAbstract\nIn this talk\, we will discuss a geom
etric construction of p-adic analogues of Maass--Shimura differential oper
ators on automorphic forms on Shimura varieties of PEL type A or C (that i
s\, unitary or symplectic)\, at p an unramified prime. Maass--Shimura oper
ators are smooth weight raising differential operators used in the study o
f special values of L-functions\, and in the arithmetic setting for the co
nstruction of p-adic L-functions. In this talk\, we will focus in particu
lar on the case of unitary groups of arbitrary signature\, when new phenom
ena arise for p non split. We will also discuss an application to the st
udy of modular mod p Galois representations. This talk is based on joint w
ork with Ellen Eischen (in the unitary case for p non split)\, and with Ei
schen\, Flanders\, Ghitza\, and Mc Andrew (in the other cases).\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Si Ying Lee (Harvard University)
DTSTART:20201118T200000Z
DTEND:20201118T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/14
DESCRIPTION:Title: Eichler-Shimura relations for Hodge type Shimura varieties\n
by Si Ying Lee (Harvard University) as part of Harvard number theory semin
ar\n\n\nAbstract\nThe well-known classical Eichler-Shimura relation for mo
dular curves asserts that the Hecke operator $T_p$ is equal\, as an algebr
aic correspondence over the special fiber\, to the sum of Frobenius and Ve
rschebung. Blasius and Rogawski proposed a generalization of this result f
or general Shimura varieties with good reduction at $p$\, and conjectured
that the Frobenius satisfies a certain Hecke polynomial. I will talk about
a recent proof of this conjecture for Shimura varieties of Hodge type\, a
ssuming a technical condition on the unramified sigma-conjugacy classes in
the associated Kottwitz set.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Loeffler (University of Warwick)
DTSTART:20201014T190000Z
DTEND:20201014T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/15
DESCRIPTION:Title: The Bloch--Kato conjecture for GSp(4)\nby David Loeffler (Un
iversity of Warwick) as part of Harvard number theory seminar\n\n\nAbstrac
t\nThe Bloch--Kato conjecture predicts that the dimension of the Selmer gr
oup of a global Galois representation is equal to the order of vanishing o
f its L-function. In this talk\, I will focus on the 4-dimensional Galois
representations arising from cohomological automorphic representations of
GSp(4) (i.e. from genus two Siegel modular forms). I will show that if the
L-function is non-vanishing at some critical value\, then the correspondi
ng Selmer group is zero\, under a long list of technical hypotheses. The p
roof of this theorem relies on an Euler system\, a p-adic L-function\, and
a reciprocity law connecting those together. I will also survey work in p
rogress aiming to extend this result to some other classes of automorphic
representations.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuya Wang (Duke University)
DTSTART:20200930T190000Z
DTEND:20200930T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/16
DESCRIPTION:Title: Pointwise Bound for $\\ell$-torsion of Class Groups\nby Jiuy
a Wang (Duke University) as part of Harvard number theory seminar\n\n\nAbs
tract\n$\\ell$-torsion conjecture states that $\\ell$-torsion of the class
group $|\\text{Cl}_K[\\ell]|$ for every number field $K$ is bounded by $\
\text{Disc}(K)^{\\epsilon}$. It follows from a classical result of Brauer-
Siegel\, or even earlier result of Minkowski that the class number $|\\tex
t{Cl}_K|$ of a number field $K$ are always bounded by $\\text{Disc}(K)^{1/
2+\\epsilon}$\, therefore we obtain a trivial bound $\\text{Disc}(K)^{1/2+
\\epsilon}$ on $|\\text{Cl}_K[\\ell]|$. We will talk about results on this
conjecture\, and recent works on breaking the trivial bound for $\\ell$-t
orsion of class groups in some cases based on a work of Ellenberg-Venkates
h.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Fintzen (Cambridge/Duke/IAS)
DTSTART:20200916T190000Z
DTEND:20200916T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/17
DESCRIPTION:Title: Representations of p-adic groups and applications\nby Jessic
a Fintzen (Cambridge/Duke/IAS) as part of Harvard number theory seminar\n\
n\nAbstract\nThe Langlands program is a far-reaching collection of conject
ures that relate different areas of mathematics including number theory an
d representation theory. A fundamental problem on the representation theor
y side of the Langlands program is the construction of all (irreducible\,
smooth\, complex) representations of p-adic groups.\n\nI will provide an o
verview of our understanding of the representations of p-adic groups\, wit
h an emphasis on recent progress.\n\nI will also outline how new results a
bout the representation theory of p-adic groups can be used to obtain cong
ruences between arbitrary automorphic forms and automorphic forms which ar
e supercuspidal at p\, which is joint work with Sug Woo Shin. This simplif
ies earlier constructions of attaching Galois representations to automorph
ic representations\, i.e. the global Langlands correspondence\, for genera
l linear groups. Moreover\, our results apply to general p-adic groups and
have therefore the potential to become widely applicable beyond the case
of the general linear group.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART:20200923T140000Z
DTEND:20200923T150000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/18
DESCRIPTION:Title: Multiplicative functions in short intervals revisited\nby Ka
isa Matomäki (University of Turku) as part of Harvard number theory semin
ar\n\n\nAbstract\nA few years ago Maksym Radziwill and I showed that the a
verage of a multiplicative function in almost all very short intervals $[x
\, x+h]$ is close to its average on a long interval $[x\, 2x]$. This resul
t has since been utilized in many applications.\nI will talk about recent
work\, where Radziwill and I revisit the problem and generalise our result
to functions which vanish often as well as prove a power-saving upper bou
nd for the number of exceptional intervals (i.e. we show that there are $O
(X/h^\\kappa)$ exceptional $x \\in [X\, 2X]$).\nWe apply this result for i
nstance to studying gaps between norm forms of an arbitrary number field.\
n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziyang Gao (CNRS/IMJ-PRG)
DTSTART:20201007T190000Z
DTEND:20201007T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/19
DESCRIPTION:Title: Bounding the number of rational points on curves\nby Ziyang
Gao (CNRS/IMJ-PRG) as part of Harvard number theory seminar\n\n\nAbstract\
nMazur conjectured\, after Faltings’s proof of the Mordell conjecture\,
that the number of rational points on a curve of genus g at least 2 define
d over a number field of degree d is bounded in terms of g\, d and the Mor
dell-Weil rank. In particular the height of the curve is not involved. In
this talk I will explain how to prove this conjecture and some generalizat
ions. I will focus on how functional transcendence and unlikely intersecti
ons are applied in the proof. If time permits\, I will talk about how the
dependence on d can be furthermore removed if we moreover assume the relat
ive Bogomolov conjecture. This is joint work with Vesselin Dimitrov and Ph
ilipp Habegger.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niki Myrto Mavraki (Harvard University)
DTSTART:20210127T200000Z
DTEND:20210127T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/20
DESCRIPTION:Title: Arithmetic dynamics of random polynomials\nby Niki Myrto Mav
raki (Harvard University) as part of Harvard number theory seminar\n\n\nAb
stract\nWe begin with an introduction to arithmetic dynamics and heights\n
attached to rational maps. We then introduce a dynamical version of Lang's
\nconjecture concerning the minimal canonical height of non-torsion ration
al\npoints in elliptic curves (due to Silverman) as well as a conjectural\
nanalogue of Mazur/Merel's theorem on uniform bounds of rational torsion\n
points in elliptic curves (due to Morton-Silverman). It is likely that the
\ntwo conjectures are harder in the dynamical setting due to the lack of\n
structure coming from a group law. We describe joint work with Pierre Le\n
Boudec in which we establish statistical versions of these conjectures for
\npolynomial maps.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timo Richarz (TU Darmstadt)
DTSTART:20210407T190000Z
DTEND:20210407T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/21
DESCRIPTION:Title: The motivic Satake equivalence\nby Timo Richarz (TU Darmstad
t) as part of Harvard number theory seminar\n\n\nAbstract\nThe geometric S
atake equivalence due to Lusztig\, Drinfeld\, Ginzburg\, Mirković and Vil
onen is an indispensable tool in the Langlands program. Versions of this e
quivalence are known for different cohomology theories such as Betti cohom
ology or algebraic D-modules over characteristic zero fields and $\\ell$-a
dic cohomology over arbitrary fields. In this talk\, I explain how to appl
y the theory of motivic complexes as developed by Voevodsky\, Ayoub\, Cisi
nski-Déglise and many others to the construction of a motivic Satake equi
valence. Under suitable cycle class maps\, it recovers the classical equiv
alence. As dual group\, one obtains a certain extension of the Langlands d
ual group by a one dimensional torus. A key step in the proof is the const
ruction of intersection motives on affine Grassmannians. A direct conseque
nce of their existence is an unconditional construction of IC-Chow groups
of moduli stacks of shtukas. My hope is to obtain on the long run independ
ence-of-$\\ell$ results in the work of V. Lafforgue on the Langlands corre
spondence for function fields. This is ongoing joint work with Jakob Schol
bach from Münster.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Scholze (University of Bonn)
DTSTART:20210203T200000Z
DTEND:20210203T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/22
DESCRIPTION:Title: Analytic geometry\nby Peter Scholze (University of Bonn) as
part of Harvard number theory seminar\n\n\nAbstract\nWe will outline a def
inition of analytic spaces that relates\nto complex- or rigid-analytic var
ieties in the same way that schemes\nrelate to algebraic varieties over a
field. Joint with Dustin Clausen.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Johansson (Chalmers/Gothenburg)
DTSTART:20210224T200000Z
DTEND:20210224T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/23
DESCRIPTION:Title: On the Calegari--Emerton conjectures for abelian type Shimura va
rieties\nby Christian Johansson (Chalmers/Gothenburg) as part of Harva
rd number theory seminar\n\n\nAbstract\nEmerton's completed cohomology giv
es\, at present\, the most general notion of a space of p-adic automorphic
forms. Important properties of completed cohomology\, such as its 'size'\
, is predicted by a conjecture of Calegari and Emerton\, which may be view
ed as a non-abelian generalization of the Leopoldt conjecture. I will disc
uss the proof of many new cases of this conjecture\, using a mixture of te
chniques from p-adic and real geometry. This is joint work with David Hans
en.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Pollack (UCSD)
DTSTART:20210317T190000Z
DTEND:20210317T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/24
DESCRIPTION:Title: Modular forms on G_2\nby Aaron Pollack (UCSD) as part of Har
vard number theory seminar\n\n\nAbstract\nFollowing work of Gross-Wallach\
, Gan-Gross-Savin defined what are called "modular forms" on the split exc
eptional group G_2. These are a special class of automorphic forms on G_2
. I'll review their definition\, and give an update about what is known
about them. Results include a construction of cuspidal modular forms with
all algebraic Fourier coefficients\, and the exact functional equation of
the completed standard L-function of certain cusp forms. The results on
L-functions are joint with Fatma Cicek\, Giuliana Davidoff\, Sarah Dijols\
, Trajan Hammonds\, and Manami Roy.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Litt (University of Georgia)
DTSTART:20210324T190000Z
DTEND:20210324T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/25
DESCRIPTION:Title: Single-valued Hodge\, p-adic^2\, and tropical integration\nb
y Daniel Litt (University of Georgia) as part of Harvard number theory sem
inar\n\n\nAbstract\nI'll discuss 4 different types of integration -- one i
n the\ncomplex setting\, one in the tropical setting\, and two in the p-ad
ic\nsetting\, and the relationships between them. In particular\, we expla
in how\nto compute Vologodsky's "single-valued" iterated integrals on curv
es of bad\nreduction in terms of Berkovich integrals\, and how to give a s
ingle-valued\nintegration theory on complex varieties. Time permitting\, I
'll explain some\npotential arithmetic applications. This is a report on j
oint work in\nprogress with Sasha Shmakov (in the complex setting) and Eri
c Katz (in the\np-adic setting).\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:François Charles (Université Paris-Sud)
DTSTART:20210414T190000Z
DTEND:20210414T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/26
DESCRIPTION:Title: Arithmetic curves lying in compact subsets of affine schemes
\nby François Charles (Université Paris-Sud) as part of Harvard number t
heory seminar\n\n\nAbstract\nWe will describe the notion of affine schemes
and their modifications in the context of Arakelov geometry. Using geomet
ry of numbers in infinite rank\, we will study their cohomological propert
ies. Concretely\, given an affine scheme X over Z and a compact subset K o
f the set of complex points of X\, we will investigate the geometry of tho
se proper arithmetic curves in X whose complex points lie in K. This is jo
int work with Jean-Benoît Bost.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bhargav Bhatt (University of Michigan)
DTSTART:20210421T190000Z
DTEND:20210421T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/27
DESCRIPTION:Title: The absolute prismatic site\nby Bhargav Bhatt (University of
Michigan) as part of Harvard number theory seminar\n\n\nAbstract\nThe abs
olute prismatic site of a p-adic formal scheme carries interesting\narithm
etic and geometric information attached to the formal scheme. In this\ntal
k\, after recalling the definition of this site\, I will discuss an\nalgeb
ro-geometric (stacky) approach to absolute prismatic cohomology and\nits c
oncomitant structures (joint with Lurie\, and partially due\nindependently
to Drinfeld). As a geometric application\, I'll explain\nDrinfeld's refin
ement of the Deligne-Illusie theorem on Hodge-to-de Rham\ndegeneration. On
the arithmetic side\, I'll describe a new classification of\ncrystalline
representations of the Galois group of a local field in terms\nof F-crysta
ls on the site (joint with Scholze).\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi Pera (Boston College)
DTSTART:20210310T200000Z
DTEND:20210310T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/28
DESCRIPTION:Title: Existence of CM lifts for points on Shimura varieties\nby Ke
erthi Madapusi Pera (Boston College) as part of Harvard number theory semi
nar\n\n\nAbstract\nI'll explain a very simple proof of the fact that K3 su
rfaces of\nfinite height admit (many) CM lifts\, a result due independentl
y to\nIto-Ito-Koshikawa and Z. Yang\, which was used by the former to prov
e the\nTate conjecture for products of K3s. This will be done directly sho
wing\nthat the deformation ring of a polarized K3 surface of finite height
admits\nas a quotient that of its Brauer group. The method applies more g
enerally\nto many isogeny classes of points on Shimura varieties of abelia
n type.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura DeMarco (Harvard University)
DTSTART:20210210T200000Z
DTEND:20210210T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/29
DESCRIPTION:Title: Elliptic surfaces\, bifurcations\, and equidistribution\nby
Laura DeMarco (Harvard University) as part of Harvard number theory semina
r\n\n\nAbstract\nIn joint work with Myrto Mavraki\, we studied the arithme
tic intersection of\nsections of elliptic surfaces\, defined over number f
ields. I will describe\nour results and formulate some related open quest
ions about families of\nmaps (dynamical systems) on P^1 over a base curve.
\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART:20210428T190000Z
DTEND:20210428T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/30
DESCRIPTION:Title: From Ramanujan to K-theory\nby Frank Calegari (University of
Chicago) as part of Harvard number theory seminar\n\n\nAbstract\nThe Roge
rs-Ramanujan identity is an equality between a certain “q-series” (giv
en as an infinite sum) and a certain modular form (given as an infinite pr
oduct). Motivated by ideas from physics\, Nahm formulated a necessary cond
ition for when such q-hypergeometric series were modular. Perhaps surprisi
ngly\, this turns out to be related to algebraic K-theory. We discuss a pr
oof of this conjecture. This is joint work with Stavros Garoufalidis and D
on Zagier.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Kühne (University of Copenhagen)
DTSTART:20210303T200000Z
DTEND:20210303T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/31
DESCRIPTION:Title: Equidistribution and Uniformity in Families of Curves\nby La
rs Kühne (University of Copenhagen) as part of Harvard number theory semi
nar\n\n\nAbstract\nIn the talk\, I will present an equidistribution result
for families of (non-degenerate) subvarieties in a (general) family of ab
elian varieties. This extends a result of DeMarco and Mavraki for curves i
n fibered products of elliptic surfaces. Using this result\, one can deduc
e a uniform version of the classical Bogomolov conjecture for curves embed
ded in their Jacobians\, namely that the number of torsion points lying on
them is uniformly bounded in the genus of the curve. This has been previo
usly only known in a few select cases by work of David--Philippon and DeMa
rco--Krieger--Ye. Finally\, one can obtain a rather uniform version of the
Mordell-Lang conjecture as well by complementing a result of Dimitrov--Ga
o--Habegger: The number of rational points on a smooth algebraic curve def
ined over a number field can be bounded solely in terms of its genus and t
he Mordell-Weil rank of its Jacobian. Again\, this was previously known on
ly under additional assumptions (Stoll\, Katz--Rabinoff--Zureick-Brown).\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (Harvard University)
DTSTART:20210217T200000Z
DTEND:20210217T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/32
DESCRIPTION:Title: Twisted derived equivalences and the Tate conjecture for K3 squa
res\nby Ziquan Yang (Harvard University) as part of Harvard number the
ory seminar\n\n\nAbstract\nThere is a long standing connection between the
Tate conjecture in codimension 1 and finiteness properties\, which first
appeared in Tate's seminal work on the endomorphisms of abelian varieties.
I will explain how one can possibly extend this connection to codimension
2 cycles\, using the theory of Brauer groups\, moduli of twisted sheaves\
, and twisted derived equivalences\, and prove the Tate conjecture for K3
squares. This recovers an earlier result of Ito-Ito-Kashikawa\, which was
established via a CM lifting theory\, and moreover provides a recipe of co
nstructing all the cycles on these varieties by purely geometric methods.\
n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melanie Matchett Wood (Harvard University)
DTSTART:20210908T190000Z
DTEND:20210908T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/33
DESCRIPTION:Title: The average size of 3-torsion in class groups of 2-extensions\nby Melanie Matchett Wood (Harvard University) as part of Harvard number
theory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbst
ract\nThe p-torsion in the class group of a number field K is conjectured
to\nbe small: of size at most $|\\operatorname{Disc} K|^\\varepsilon$\, an
d to have constant\naverage size in families with a given Galois closure g
roup (when p\ndoesn't divide the order of the group). In general\, the be
st upper\nbound we have is $|\\operatorname{Disc} K|^{1/2+\\varepsilon}$\,
and previously the only two\ncases known with constant average were for 3
-torsion in quadratic\nfields (Davenport and Heilbronn\, 1971) and 2-torsi
on in non-Galois\ncubic fields (Bhargava\, 2005). We prove that the 3-tor
sion is\nconstant on average for fields with Galois closure group any 2-gr
oup\nwith a transposition\, including\, e.g. quartic $D_4$ fields. We wil
l\ndiscuss the main inputs into the proof with an eye towards giving an\ni
ntroduction to the tools in the area. This is joint work with Robert\nLem
ke Oliver and Jiuya Wang.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (Harvard University)
DTSTART:20210929T190000Z
DTEND:20210929T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/34
DESCRIPTION:Title: Density of arithmetic Hodge loci\nby Salim Tayou (Harvard Un
iversity) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nI will explain a conjecture on de
nsity of arithmetic Hodge loci which includes and generalizes several rece
nt density results of these loci in arithmetic geometry. This conjecture h
as also analogues over functions fields that I will survey. As a particula
r instance\, I will outline the proof of the following result: a K3 surfac
e over a number field admits infinitely many specializations where its Pic
ard rank jumps. This last result is joint work with Ananth Shankar\, Arul
Shankar and Yunqing Tang.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Kieffer (Harvard University)
DTSTART:20211006T190000Z
DTEND:20211006T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/35
DESCRIPTION:Title: Higher-dimensional modular equations and point counting on abeli
an surfaces\nby Jean Kieffer (Harvard University) as part of Harvard n
umber theory seminar\n\nLecture held in Room 507 in the Science Center.\n\
nAbstract\nGiven a prime number l\, the elliptic modular polynomial of lev
el l is an explicit equation for the locus of elliptic curves related by a
n l-isogeny. These polynomials have a large number of algorithmic applicat
ions: in particular\, they are an essential ingredient in the celebrated S
EA algorithm for counting points on elliptic curves over finite fields of
large characteristic.\n\nMore generally\, modular equations describe the l
ocus of isogenous abelian varieties in certain moduli spaces called PEL Sh
imura varieties. We will present upper bounds on the size of modular equat
ions in terms of their level\, and outline a general strategy to compute a
n isogeny A->A' given a point (A\,A') where the modular equations are sati
sfied. This generalizes well-known properties of elliptic modular polynomi
als to higher dimensions.\n\nThe isogeny algorithm is made fully explicit
for Jacobians of genus 2 curves. In combination with efficient evaluation
methods for modular equations in genus 2 via complex approximations\, this
gives rise to point counting algorithms for (Jacobians of) genus 2 curves
whose\nasymptotic costs\, under standard heuristics\, improve on previous
results.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard University)
DTSTART:20210915T190000Z
DTEND:20210915T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/36
DESCRIPTION:Title: Finitely Presented Groups in Arithmetic Geometry\nby Mark Sh
usterman (Harvard University) as part of Harvard number theory seminar\n\n
Lecture held in Room 507 in the Science Center.\n\nAbstract\nWe discuss th
e problem of determining the number of generators and relations of several
profinite groups of arithmetic and geometric origin. \nThese include etal
e fundamental groups of smooth projective varieties\, absolute Galois grou
ps of local fields\, and Galois groups of maximal unramified extensions of
number fields. The results are based on a cohomological presentability cr
iterion of Lubotzky\, and draw inspiration from well-known facts about thr
ee-dimensional manifolds\, as in arithmetic topology. \n\nThe talk is ba
sed on a joint work with Esnault and Srinivas\, on a joint work with Jarde
n\, and on work of Yuan Liu.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (Harvard University)
DTSTART:20210922T190000Z
DTEND:20210922T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/37
DESCRIPTION:Title: Galois action on the pro-algebraic completion of the fundamental
group\nby Alexander Petrov (Harvard University) as part of Harvard nu
mber theory seminar\n\nLecture held in Room 507 in the Science Center.\n\n
Abstract\nGiven a variety over a number field\, its geometric etale\nfunda
mental group comes equipped with an action of the Galois group. This\nindu
ces a Galois action on the pro-algebraic completion of the etale\nfundamen
tal group and hence the ring of functions on that pro-algebraic\ncompletio
n provides a supply of Galois representations.\n\nIt turns out that any fi
nite-dimensional p-adic Galois representation\ncontained in the ring of fu
nctions on the pro-algebraic completion of the\nfundamental group of a smo
oth variety satisfies the assumptions of the\nFontaine-Mazur conjecture: i
t is de Rham at places above p and is a. e.\nunramified.\n\nConversely\, w
e will show that every semi-simple representation of the\nGalois group of
a number field coming from algebraic geometry (that is\,\nappearing as a s
ubquotient of the etale cohomology of an algebraic variety)\ncan be establ
ished as a subquotient of the ring of functions on the\npro-algebraic comp
letion of the fundamental group of the projective line\nwith 3 punctures.\
n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard University)
DTSTART:20211020T190000Z
DTEND:20211020T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/38
DESCRIPTION:Title: Effective height bounds for odd-degree totally real points on so
me curves\nby Levent Alpöge (Harvard University) as part of Harvard n
umber theory seminar\n\nLecture held in Room 507 in the Science Center.\n\
nAbstract\nI will give a finite-time algorithm that\, on input (g\,K\,S) w
ith g > 0\, K a totally real number field of odd degree\, and S a finite s
et of places of K\, outputs the finitely many g-dimensional abelian variet
ies A/K which are of $\\operatorname{GL}_2$-type over K and have good redu
ction outside S.\n\nThe point of this is to effectively compute the S-inte
gral K-points on a Hilbert modular variety\, and the point of that is to b
e able to compute all K-rational points on complete curves inside such var
ieties.\n\nThis gives effective height bounds for rational points on infin
itely many curves and (for each curve) over infinitely many number fields.
For example one gets effective height points for odd-degree totally real
points on $x^6 + 4y^3 = 1$\, by using the hypergeometric family associated
to the arithmetic triangle group $\\Delta(3\,6\,6)$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Zhang (MIT)
DTSTART:20211027T190000Z
DTEND:20211027T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/39
DESCRIPTION:Title: p-adic Heights of the arithmetic diagonal cycles\nby Wei Zha
ng (MIT) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nThis is a work in progress joint
with Daniel Disegni. We formulate a p-adic analogue of the Arithmetic Gan-
-Gross--Prasad conjecture for unitary groups\, relating the p-adic height
pairing of the arithmetic diagonal cycles to the first central derivative
(along the cyclotomic direction) of a p-adic Rankin—Selberg L-function
associated to cuspidal automorphic representations. In the good ordinary c
ase we are able to prove the conjecture\, at least when the ramifications
are mild at inert primes. We deduce some application to the p-adic version
of the Bloch-Kato conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiwei Yun (MIT)
DTSTART:20211103T190000Z
DTEND:20211103T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/40
DESCRIPTION:Title: Special cycles for unitary Shtukas and modularity\nby Zhiwei
Yun (MIT) as part of Harvard number theory seminar\n\nLecture held in Roo
m 507 in the Science Center.\n\nAbstract\nWe define a generating series of
algebraic cycles on the moduli\nstack of unitary Shtukas and conjecture t
hat it is a Chow-group valued\nautomorphic form. This is a function field
analogue of the special cycles\ndefined by Kudla and Rapoport\, but with a
n extra degree of freedom namely\nthe number of legs of the Shtukas. I wil
l talk about several pieces of\nevidence for the conjecture. This is joint
work with Tony Feng and Wei\nZhang.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT)
DTSTART:20211110T200000Z
DTEND:20211110T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/41
DESCRIPTION:Title: The Galois action on symplectic K-theory\nby Tony Feng (MIT)
as part of Harvard number theory seminar\n\nLecture held in Room 507 in t
he Science Center.\n\nAbstract\nA phenomenon underlying many remarkable re
sults in number theory is the natural Galois action on the cohomology of s
ymplectic groups of integers. In joint work with Soren Galatius and Akshay
Venkatesh\, we define a symplectic variant of algebraic K-theory\, which
carries a natural Galois action for similar reasons. We compute this Galoi
s action and characterize it in terms of a universality property\, in the
spirit of the Langlands philosophy.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siyan Daniel Li-Huerta (Harvard University)
DTSTART:20211013T190000Z
DTEND:20211013T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/42
DESCRIPTION:Title: The plectic conjecture over local fields\nby Siyan Daniel Li
-Huerta (Harvard University) as part of Harvard number theory seminar\n\nL
ecture held in Room 507 in the Science Center.\n\nAbstract\nThe étale coh
omology of varieties over Q enjoys a Galois action. In the\ncase of Hilber
t modular varieties\, Nekovář-Scholl observed that this Galois\naction o
n the level of cohomology extends to a much larger profinite group:\nthe p
lectic group. They conjectured that this extension holds even on the\nleve
l of complexes\, as well as for more general Shimura varieties.\n\nWe pres
ent a proof of the analogue of this conjecture for local Shimura\nvarietie
s. This includes (the generic fibers of) Lubin–Tate spaces\,\nDrinfeld u
pper half spaces\, and more generally Rapoport–Zink spaces. The\nproof c
rucially uses Scholze's theory of diamonds.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Howard (Boston College)
DTSTART:20211117T200000Z
DTEND:20211117T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/43
DESCRIPTION:Title: Arithmetic volumes of unitary Shimura varieties\nby Benjamin
Howard (Boston College) as part of Harvard number theory seminar\n\nLectu
re held in Room 507 in the Science Center.\n\nAbstract\nThe integral model
of a GU(n-1\,1) Shimura variety carries a natural metrized line bundle of
modular forms. Viewing this metrized line bundle as a class in the codim
ension one arithmetic Chow group\, one can define its arithmetic volume as
an iterated self-intersection. We will show that this volume can be expr
essed in terms of logarithmic derivatives of Dirichlet L-functions at inte
ger points\, and explain the connection with the arithmetic Siegel-Weil co
njecture of Kudla-Rapoport. This is joint work with Jan Bruinier.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Kisin (Harvard University)
DTSTART:20211201T200000Z
DTEND:20211201T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/44
DESCRIPTION:Title: Mod p points on Shimura varieties\nby Mark Kisin (Harvard Un
iversity) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nThe study of mod p points on Shim
ura varieties was originally\nmotivated by the study of the Hasse-Weil zet
a function for Shimura\nvarieties.\nIt involves some rather subtle problem
s which test just how much we know\nabout motives over finite fields. In t
his talk I will explain some recent\nresults\, and\napplications\, and wha
t still remains conjectural.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Betts (Harvard University)
DTSTART:20220209T200000Z
DTEND:20220209T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/45
DESCRIPTION:Title: Galois sections and the method of Lawrence--Venkatesh\nby Al
exander Betts (Harvard University) as part of Harvard number theory semina
r\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nGrothend
ieck's Section Conjecture posits that the set of rational\npoints on a smo
oth projective curve Y of genus at least two should be equal\nto a certain
"section set" defined purely in terms of the etale fundamental\ngroup of
Y. In this talk\, I will preview some upcoming work with Jakob Stix\nin wh
ich we prove a partial finiteness result for this section set\, thereby\ng
iving an unconditional verification of a prediction of the Section\nConjec
ture for a general curve Y. We do this by adapting the recent p-adic\nproo
f of the Mordell Conjecture due to Brian Lawrence and Akshay Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fabian Gundlach (Harvard University)
DTSTART:20220202T200000Z
DTEND:20220202T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/46
DESCRIPTION:Title: Counting quaternionic extensions\nby Fabian Gundlach (Harvar
d University) as part of Harvard number theory seminar\n\nLecture held in
Room 507 in the Science Center.\n\nAbstract\nConsider the set of Galois ex
tensions $L$ of $\\mathbb Q$ whose Galois group is the quaternion group. F
or large $X$\, Klüners counted extensions with $|\\mathrm{disc}(L)| <= X$
. We discuss asymptotics when bounding invariants other than the discrimin
ant.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard University)
DTSTART:20220216T200000Z
DTEND:20220216T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/47
DESCRIPTION:Title: Kolyvagin's conjecture\, bipartite Euler systems\, and higher co
ngruences of modular forms\nby Naomi Sweeting (Harvard University) as
part of Harvard number theory seminar\n\nLecture held in Room 507 in the S
cience Center.\n\nAbstract\nFor an elliptic curve E\, Kolyvagin used Heeg
ner points to construct\nspecial Galois cohomology classes valued in the t
orsion points of E. Under\nthe conjecture that not all of these classes va
nish\, he showed that they\nencode the Selmer rank of E. I will explain a
proof of new cases of this\nconjecture that builds on prior work of Wei Zh
ang. The proof naturally\nleads to a generalization of Kolyvagin's work in
a complimentary "definite"\nsetting\, where Heegner points are replaced b
y special values of a\nquaternionic modular form. I'll also explain an "ul
trapatching" formalism\nwhich simplifies the Selmer group arguments requir
ed for the proof.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Harvard University)
DTSTART:20220223T200000Z
DTEND:20220223T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/48
DESCRIPTION:Title: Geometric local systems on very general curves\nby Aaron Lan
desman (Harvard University) as part of Harvard number theory seminar\n\nLe
cture held in Room 507 in the Science Center.\n\nAbstract\nConjectures of
Esnault-Kerz and Budur-Wang state\nthat the locus of rank r complex local
systems on a complex variety\nof geometric origin are Zariski dense in the
character variety\nparameterizing complex rank r local systems.\nIn joint
work with Daniel Litt\, we show these conjectures fail to hold when\nX is
a sufficiently general curve of genus $g$ and $r < 2\\sqrt{g+1}$\nby show
ing that any such local system coming from geometry is in fact\nisotrivial
.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Myrto Mavraki (Harvard University)
DTSTART:20220302T200000Z
DTEND:20220302T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/49
DESCRIPTION:Title: Towards uniformity in the dynamical Bogomolov conjecture\nby
Myrto Mavraki (Harvard University) as part of Harvard number theory semin
ar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nInspire
d by an analogy between torsion and preperiodic points\,\nZhang has propos
ed a dynamical generalization of the classical\nManin-Mumford and Bogomolo
v conjectures. A special case of these\nconjectures\, for `split' maps\, h
as recently been established by Nguyen\,\nGhioca and Ye. In particular\, t
hey show that two rational maps have at most\nfinitely many common preperi
odic points\, unless they are `related'. Recent\nbreakthroughs by Dimitrov
\, Gao\, Habegger and Kühne have established that\nthe classical Bogomolo
v conjecture holds uniformly across curves of given\ngenus.\nIn this talk
we discuss uniform versions of the dynamical Bogomolov\nconjecture across
1-parameter families of split maps and curves. To this\nend\, we establish
instances of a 'relative dynamical Bogomolov conjecture'.\nThis is joint
work with Harry Schmidt (University of Basel).\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Pollack (Boston University)
DTSTART:20220427T190000Z
DTEND:20220427T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/50
DESCRIPTION:Title: Slopes of modular forms and reductions of crystalline representa
tions\nby Robert Pollack (Boston University) as part of Harvard number
theory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbst
ract\nThe ghost conjecture predicts slopes of modular forms whose\nresidua
l representation is locally reducible. In this talk\, we'll examine\nloca
lly irreducible representations and discuss recent progress on\nformulatin
g a conjecture in this case. It's a lot trickier and the story\nremains i
ncomplete\, but we will discuss how an irregular ghost conjecture\nis inti
mately related to reductions of crystalline representations.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART:20220420T190000Z
DTEND:20220420T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/51
DESCRIPTION:Title: Quadratic Chabauty for modular curves\nby Jennifer Balakrish
nan (Boston University) as part of Harvard number theory seminar\n\nLectur
e held in Room 507 in the Science Center.\n\nAbstract\nAbstract: We descri
be how p-adic height pairings can be used to\ndetermine the set of rationa
l points on curves\, in the spirit of Kim's\nnonabelian Chabauty program.
In particular\, we discuss what aspects of\nthe quadratic Chabauty method
can be made practical for certain\nmodular curves. This is joint work with
Netan Dogra\, Steffen Mueller\,\nJan Tuitman\, and Jan Vonk.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allechar Serrano López
DTSTART:20220309T200000Z
DTEND:20220309T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/52
DESCRIPTION:Title: Counting fields generated by points on plane curves\nby Alle
char Serrano López as part of Harvard number theory seminar\n\nLecture he
ld in Room 507 in the Science Center.\n\nAbstract\nFor a smooth projective
curve $C/\\mathbb{Q}$\, how many field\nextensions of $\\mathbb{Q}$ -- of
given degree and bounded discriminant ---\narise from adjoining a point o
f $C(\\overline{\\mathbb{Q}})$? Can we further\ncount the number of such e
xtensions with a specified Galois group?\nAsymptotic lower bounds for thes
e quantities have been found for elliptic\ncurves by Lemke Oliver and Thor
ne\, for hyperelliptic curves by Keyes\, and\nfor superelliptic curves by
Beneish and Keyes. We discuss similar\nasymptotic lower bounds that hold f
or all smooth plane curves $C$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia University)
DTSTART:20220323T190000Z
DTEND:20220323T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/53
DESCRIPTION:Title: A visit to 3-manifolds in the quest to understand random Galois
groups\nby Will Sawin (Columbia University) as part of Harvard number
theory seminar\n\nLecture held in Room 507 in the Science Center.\n\nAbstr
act\nCohen and Lenstra gave a conjectural distribution for the class group
of a random quadratic number field. Since the class group is the Galois g
roup of the maximum abelian unramified extension\, a natural generalizatio
n would be to give a conjecture for the distribution of the Galois group o
f the maximal unramified extension. Previous work has produced a plausible
conjecture in special cases\, with the most general being recent work of
Liu\, Wood\, and Zurieck-Brown.\n\nThere is a deep analogy between number
fields and 3-manifolds. Thus\, an analogous question would be to describe
the distribution of the profinite completion of the fundamental group of a
random 3-manifold. In this talk\, I will explain how Melanie Wood and I a
nswered this question for a model of random 3-manifolds defined by Dunfiel
d and Thurston\, and how the techniques we used should allow us\, in futur
e work\, to give a more general conjecture in the number field case.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Urban (Columbia University)
DTSTART:20220413T190000Z
DTEND:20220413T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/54
DESCRIPTION:Title: Euler systems and the p-adic Langlands correspondence\nby Er
ic Urban (Columbia University) as part of Harvard number theory seminar\n\
nLecture held in Room 507 in the Science Center.\n\nAbstract\nAbout 2 year
s ago\, I have given a new construction of the Euler system of cyclotomic
units via Eisenstein congruences in which the p-adic Langlands correspond
ence for $\\GL_2(\\Q_p)$ plays a central role. In this talk\, I want to ex
plain how one can extend this method to obtain a large class of new Euler
systems attached to ordinary automorphic forms. This is a work in progress
.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (Princeton University)
DTSTART:20220330T190000Z
DTEND:20220330T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/55
DESCRIPTION:Title: The unbounded denominators conjecture\nby Yunqing Tang (Prin
ceton University) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nThe unbounded denominator
s conjecture\, first raised by Atkin and Swinnerton-Dyer\, asserts that a
modular form for a finite index subgroup of $\\SL_2(\\mathbb Z)$ whose Fou
rier coefficients have bounded denominators must be a modular form for som
e congruence subgroup. In this talk\, we will give a sketch of the proof o
f this conjecture based on a new arithmetic algebraization theorem. This i
s joint work with Frank Calegari and Vesselin Dimitrov.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Baker (Georgia Institute of Technology)
DTSTART:20220504T190000Z
DTEND:20220504T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/56
DESCRIPTION:Title: Non-archimedean and tropical geometry\, algebraic groups\, modul
i spaces of matroids\, and the field with one element\nby Matt Baker (
Georgia Institute of Technology) as part of Harvard number theory seminar\
n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nI will giv
e an introduction to Oliver Lorscheid’s theory of\nordered blueprints
– one of the more successful approaches to “the field of\none element
” – and sketch its relationship to Berkovich spaces\, tropical\ngeomet
ry\, Tits models for algebraic groups\, and moduli spaces of matroids.\nTh
e basic idea for the latter two applications is quite simple: given a\nsch
eme over Z defined by equations with coefficients in {0\,1\,-1}\, t
here\nis a corresponding “blue model” whose K-points (where
K is the Krasner\nhyperfield) sometimes correspond to interesting comb
inatorial structures.\nFor example\, taking K-points of a suitable
blue model for a split\nreductive group scheme G over Z gives the W
eyl group of G\, and\ntaking K-points\nof a suitable blue model for
the Grassmannian G(r\,n) gives the set of\nmatroids of rank r on {1\,…\
,n}. Similarly\, the Berkovich analytification of\na scheme X over a value
d field K coincides\, as a topological space\, with\nthe set of T-p
oints of X\, considered as an ordered blue scheme over K.\nHere T i
s the tropical hyperfield\, and T-points are defined using the\nobs
ervation that a (height 1) valuation on K is nothing other than a\nhomomor
phism to T.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART:20220406T190000Z
DTEND:20220406T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/57
DESCRIPTION:Title: Isolated points on modular curves\nby Bianca Viray (Universi
ty of Washington) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nLet C be an algebraic cur
ve over a number field. Faltings's theorem on\nrational points on subvarie
ties of abelian varieties implies that all\nalgebraic points on C arise in
algebraic families\, with finitely many\nexceptions. These exceptions ar
e known as isolated points. We study how\nisolated points behave under mor
phisms and then specialize to the case of\nmodular curves. We show that i
solated points on X_1(n) push down to\nisolated points on a modular curve
whose level is bounded by a constant\nthat depends only on the j-invariant
of the isolated point. This is joint\nwork with A. Bourdon\, O. Ejder\,
Y. Liu\, and F. Odumodu.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshay Venkatesh (IAS)
DTSTART:20220914T190000Z
DTEND:20220914T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/60
DESCRIPTION:Title: Symplectic Reidemeister torsion and symplectic $L$-functions
\nby Akshay Venkatesh (IAS) as part of Harvard number theory seminar\n\nLe
cture held in Room 507 in the Science Center.\n\nAbstract\nMany of the qua
ntities appearing in the conjecture of Birch and Swinnerton-Dyer look susp
iciously like squares. Motivated by this and related examples\, we may ask
if the central value of an $L$-function "of symplectic type" admits a pre
ferred square root.\n\nThe answer is no: there's an interesting cohomologi
cal obstruction. More formally\, in the everywhere unramified situation ov
er a function field\, I will describe an explicit cohomological formula fo
r the $L$-function modulo squares. This is based on a purely topological r
esult about $3$-manifolds. If time permits I'll speculate on generalizatio
ns. This is based on joint work with Amina Abdurrahman.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (University of Michigan)
DTSTART:20220921T190000Z
DTEND:20220921T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/61
DESCRIPTION:Title: The negative Pell equation and applications\nby Peter Koyman
s (University of Michigan) as part of Harvard number theory seminar\n\nLec
ture held in Room 507 in the Science Center.\n\nAbstract\nIn this talk we
will study the negative Pell equation\, which is the conic $C_D : x^2 - D
y^2 = -1$ to be solved in integers $x\, y \\in \\mathbb{Z}$. We shall be
concerned with the following question: as we vary over squarefree integers
$D$\, how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic f
ormula for such $D$. Fouvry and Klüners gave upper and lower bounds of th
e correct order of magnitude. We will discuss a proof of Stevenhagen's con
jecture\, and potential applications of the new proof techniques. This is
joint work with Carlo Pagano.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziquan Yang (UW Madison)
DTSTART:20220928T190000Z
DTEND:20220928T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/62
DESCRIPTION:Title: The Tate conjecture for $h^{2\, 0} = 1$ varieties over finite fi
elds\nby Ziquan Yang (UW Madison) as part of Harvard number theory sem
inar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nThe p
ast decade has witnessed a great advancement on the Tate conjecture for va
rieties with Hodge number $h^{2\, 0} = 1$. Charles\, Madapusi-Pera and Mau
lik completely settled the conjecture for K3 surfaces over finite fields\,
and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for m
ore or less arbitrary $h^{2\, 0} = 1$ varieties in characteristic $0$.\n\n
In this talk\, I will explain that the Tate conjecture is true for mod $p$
reductions of complex projective $h^{2\, 0} = 1$ varieties when $p$ is bi
g enough\, under a mild assumption on moduli. By refining this general res
ult\, we prove that in characteristic $p$ at least $5$ the BSD conjecture
holds for a height $1$ elliptic curve $E$ over a function field of genus $
1$\, as long as $E$ is subject to the generic condition that all singular
fibers in its minimal compactification are irreducible. We also prove the
Tate conjecture over finite fields for a class of surfaces of general type
and a class of Fano varieties. The overall philosophy is that the connect
ion between the Tate conjecture over finite fields and the Lefschetz $(1\,
1)$-theorem over the complex numbers is very robust for $h^{2\, 0} = 1$ v
arieties\, and works well beyond the hyperkähler world.\n\nThis is based
on joint work with Paul Hamacher and Xiaolei Zhao.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Li-Huerta (Harvard)
DTSTART:20221005T190000Z
DTEND:20221005T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/63
DESCRIPTION:Title: Local-global compatibility over function fields\nby Daniel L
i-Huerta (Harvard) as part of Harvard number theory seminar\n\nLecture hel
d in Room 507 in the Science Center.\n\nAbstract\nThe Langlands program pr
edicts a relationship between automorphic representations of a reductive g
roup $G$ and Galois representations valued in its $L$-group. For general $
G$ over a global function field\, the automorphic-to-Galois direction has
been constructed by V. Lafforgue. More recently\, for general $G$ over a n
onarchimedean local field\, a similar correspondence has been constructed
by Fargues–Scholze.\n\nWe present a proof that the V. Lafforgue and Farg
ues–Scholze correspondences are compatible\, generalizing local-global c
ompatibility from class field theory. As a consequence\, the correspondenc
es of Genestier–Lafforgue and Fargues–Scholze agree\, which answers a
question of Fargues–Scholze\, Hansen\, Harris\, and Kaletha.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (Freie Universität Berlin)
DTSTART:20221012T190000Z
DTEND:20221012T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/64
DESCRIPTION:Title: Integrality properties of the Betti moduli space\nby Hélèn
e Esnault (Freie Universität Berlin) as part of Harvard number theory sem
inar\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nWe st
udy them\, in particular showing on a smooth complex quasi-projective vari
ety the existence of $\\ell$-adic absolutely irreducible local systems fo
r all $\\ell$ the moment there is a complex irreducible topological local
system. The proof is purely arithmetic.\n\nThis is work in progress with
Johan de Jong\, relying in part on earlier work with Michael Groechenig.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga (Princeton)
DTSTART:20221019T190000Z
DTEND:20221019T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/65
DESCRIPTION:Title: Arithmetic statistics via graded Lie algebras\nby Jef Laga (
Princeton) as part of Harvard number theory seminar\n\nLecture held in Roo
m 507 in the Science Center.\n\nAbstract\nI will explain how various resul
ts in arithmetic statistics by Bhargava\, Gross\, Shankar and others on $2
$-Selmer groups of Jacobians of (hyper)elliptic curves can be organised an
d reproved using the theory of graded Lie algebras\, following earlier wor
k of Thorne. This gives a uniform proof of these results and yields new th
eorems for certain families of non-hyperelliptic curves. I will also menti
on some applications to rational points on certain families of curves.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shai Haran (Technion)
DTSTART:20221026T190000Z
DTEND:20221026T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/66
DESCRIPTION:Title: Non additive geometry and Frobenius correspondences\nby Shai
Haran (Technion) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nThe usual language of alg
ebraic geometry is not appropriate for arithmetical geometry: addition is
singular at the real prime. We developed two languages that overcome this
problem: one replace s rings by the collection of “vectors” or by bi-o
perads\, and another based on “matrices” or props. Once one understand
s the delicate commutativity condition one can proceed following Grothendi
eck's footsteps exactly. The props\, when viewed up to conjugation\, give
us new commutative rings with Frobenius endomorphisms.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spencer Leslie (Boston College)
DTSTART:20221102T190000Z
DTEND:20221102T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/67
DESCRIPTION:Title: Endoscopy for symmetric varieties\nby Spencer Leslie (Boston
College) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nRelative trace formulas are centr
al tools in the study of relative functoriality. In many cases of interest
\, basic stability problems have not previously been addressed. In this ta
lk\, I discuss a theory of endoscopy in the context of symmetric varieties
with the global goal of stabilizing the associated relative trace formula
. I outline how\, using the dual group of the symmetric variety\, one can
give a good notion of endoscopic symmetric variety and conjecture a matchi
ng of relative orbital integrals in order to stabilize the relative trace
formula\, which can be proved in some cases. Time permitting\, I will expl
ain my proof of these conjectures in the case of unitary Friedberg–Jacqu
et periods.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gyujin Oh (Columbia)
DTSTART:20221109T200000Z
DTEND:20221109T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/68
DESCRIPTION:Title: Cohomological degree-shifting operators on Shimura varieties
\nby Gyujin Oh (Columbia) as part of Harvard number theory seminar\n\nLect
ure held in Room 507 in the Science Center.\n\nAbstract\nAn automorphic fo
rm can appear in multiple degrees of the cohomology of arithmetic manifold
s\, and this happens mostly when the arithmetic manifolds are not algebrai
c. This phenomenon is a part of the "derived" structures of the Langlands
program\, suggested by Venkatesh. However\, even over algebraic arithmetic
manifolds\, certain automorphic forms like weight-one elliptic modular fo
rms possess a derived structure. In this talk\, we discuss this idea over
Shimura varieties. A part of the story is the construction of archimedean/
p-adic "derived" operators on the cohomology of Shimura varieties\, using
complex/p-adic Hodge theory.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tasho Kaletha (University of Michigan)
DTSTART:20221116T200000Z
DTEND:20221116T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/69
DESCRIPTION:Title: Covers of reductive groups and functoriality\nby Tasho Kalet
ha (University of Michigan) as part of Harvard number theory seminar\n\nLe
cture held in Room 507 in the Science Center.\n\nAbstract\nTo a connected
reductive group $G$ over a local field $F$ we define a compact topological
group $\\tilde\\pi_1(G)$ and an extension $G(F)_\\infty$ of $G(F)$ by $\\
tilde\\pi_1(G)$. From any character $x$ of $\\tilde\\pi_1(G)$ of order $n$
we obtain an $n$-fold cover $G(F)_x$ of the topological group $G(F)$. We
also define an $L$-group for $G(F)_x$\, which is a usually non-split exten
sion of the Galois group by the dual group of G\, and deduce from the line
ar case a refined local Langlands correspondence between genuine represent
ations of $G(F)_x$ and $L$-parameters valued in this $L$-group.\n\nThis co
nstruction is motivated by Langlands functoriality. We show that a subgrou
p of the $L$-group of $G$ of a certain kind naturally lead to a smaller qu
asi-split group $H$ and a double cover of $H(F)$. Genuine representations
of this double cover are expected to be in functorial relationship with re
presentations of $G(F)$. We will present two concrete applications of this
\, one that gives a characterization of the local Langlands correspondence
for supercuspidal $L$-parameters when $p$ is sufficiently large\, and one
to the theory of endoscopy.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhishek Oswal (Caltech)
DTSTART:20221130T200000Z
DTEND:20221130T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/70
DESCRIPTION:Title: A $p$-adic analogue of an algebraization theorem of Borel\nb
y Abhishek Oswal (Caltech) as part of Harvard number theory seminar\n\nLec
ture held in Room 507 in the Science Center.\n\nAbstract\nLet $S$ be a Shi
mura variety such that the connected components of the set of complex poin
ts $S(\\mathbb{C})$ are of the form $D/\\Gamma$\, where $\\Gamma$ is a tor
sion-free arithmetic group acting on the Hermitian symmetric domain $D$. B
orel proved that any holomorphic map from any complex algebraic variety in
to $S(\\mathbb{C})$ is an algebraic map. In this talk I shall describe ong
oing joint work with Ananth Shankar and Xinwen Zhu\, where we prove a $p$-
adic analogue of this result of Borel for compact Shimura varieties of abe
lian type.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia)
DTSTART:20221207T200000Z
DTEND:20221207T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/71
DESCRIPTION:Title: Malle's conjecture for nilpotent groups\nby Carlo Pagano (Co
ncordia) as part of Harvard number theory seminar\n\nLecture held in Room
507 in the Science Center.\n\nAbstract\nMalle's conjecture is a quantitati
ve version of the Galois inverse problem. Namely\, fixing some ramificatio
n invariant of number fields (discriminant\, product of ramified primes\,
etc)\, for a finite group $G$ one seeks an asymptotic formula for the numb
er of $G$-extensions (of a given number field) having bounded ramification
invariant. In this talk I will overview past and ongoing joint work with
Peter Koymans focusing on the case of nilpotent groups.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Shnidman (Hebrew University of Jerusalem)
DTSTART:20230201T200000Z
DTEND:20230201T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/72
DESCRIPTION:Title: Bielliptic Picard curves\nby Ari Shnidman (Hebrew University
of Jerusalem) as part of Harvard number theory seminar\n\nLecture held in
Room 507 in the Science Center.\n\nAbstract\nI'll describe the geometry a
nd arithmetic of the curves $y^3 = x^4 + ax^2 + b$. The Jacobians of these
curves factor as a product of an elliptic curve and an abelian surface $A
$. The latter is an example of a "false elliptic curve"\, i.e. an abelian
surface with quaternionic multiplication. I'll explain how to see this fr
om the geometry of the curve\, and then I'll give some results on the Mord
ell–Weil groups $A(\\mathbb{Q})$. This is based on joint work with Laga
and Laga–Schembri–Voight.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART:20230208T200000Z
DTEND:20230208T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/73
DESCRIPTION:Title: Higher modularity of elliptic curves\nby Jared Weinstein (Bo
ston University) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nElliptic curves $E$ over t
he rational numbers are modular: this means there is a nonconstant map fro
m a modular curve to $E$. When instead the coefficients of $E$ belong to a
function field\, it still makes sense to talk about the modularity of $E$
(and this is known)\, but one can also extend the idea further and ask wh
ether $E$ is '$r$-modular' for $r=2\,3\\ldots$. To define this generalizat
ion\, the modular curve gets replaced with Drinfeld's concept of a 'shtuka
space'. The $r$-modularity of $E$ is predicted by Tate's conjecture. In j
oint work with Adam Logan\, we give some classes of elliptic curves $E$ wh
ich are $2$- and $3$-modular.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard)
DTSTART:20230215T200000Z
DTEND:20230215T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/74
DESCRIPTION:Title: Integers which are(n’t) the sum of two cubes\nby Levent Al
pöge (Harvard) as part of Harvard number theory seminar\n\nLecture held i
n Room 507 in the Science Center.\n\nAbstract\nFermat identified the integ
ers which are a sum of two squares\, integral or rational: they are exactl
y those integers which have all primes congruent to 3 (mod 4) occurring to
an even power in their prime factorization — a condition satisfied by 0
% of integers!\n\nWhat about the integers which are a sum of two cubes? 0%
are a sum of two integral cubes\, but...\n\nMain Theorem:\n\n1. A positiv
e proportion of integers aren’t the sum of two rational cubes\,\n\n2. an
d also a positive proportion are!\n\n(Joint with Manjul Bhargava and Ari S
hnidman.)\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (MIT)
DTSTART:20230222T200000Z
DTEND:20230222T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/75
DESCRIPTION:Title: Hecke algebras for $p$-adic groups and the explicit local Langla
nds correspondence for $\\mathrm{G}_2$\nby Yujie Xu (MIT) as part of H
arvard number theory seminar\n\nLecture held in Room 507 in the Science Ce
nter.\n\nAbstract\nI will talk about my recent joint work with Aubert wher
e we prove the local Langlands conjecture for $\\mathrm{G}_2$ (explicitly)
. This uses our earlier results on Hecke algebras attached to Bernstein co
mponents of reductive $p$-adic groups\, as well as an expected property on
cuspidal support\, along with a list of characterizing properties. In par
ticular\, we obtain "mixed" $L$-packets containing $F$-singular supercuspi
dals and non-supercuspidals.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Swaminathan (Harvard)
DTSTART:20230301T200000Z
DTEND:20230301T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/76
DESCRIPTION:Title: Counting integral points on symmetric varieties\, and applicatio
ns to arithmetic statistics\nby Ashvin Swaminathan (Harvard) as part o
f Harvard number theory seminar\n\nLecture held in Room 507 in the Science
Center.\n\nAbstract\nOver the past few decades\, significant progress has
been made in arithmetic statistics by the following two-step process: (1)
parametrize arithmetic objects of interest in terms of the integral orbit
s of a representation of a group $G$ acting on a vector space $V$\; and (2
) use geometry-of-numbers methods to count the orbits of $G(\\mathbb{Z})$
on $V(\\mathbb{Z})$. But it often happens that the arithmetic objects of i
nterest correspond to orbits that lie on a proper subvariety of $V$. In su
ch cases\, geometry-of-numbers methods do not suffice to obtain precise as
ymptotics\, and more sophisticated point-counting techniques are required.
In this talk\, we explain how the Eskin–McMullen method for counting in
tegral points on symmetric varieties can be used to study the distribution
of $2$-class groups in certain thin families of cubic number fields.\n\n(
Joint with Iman Setayesh\, Arul Shankar\, and Artane Siad)\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard)
DTSTART:20230308T200000Z
DTEND:20230308T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/77
DESCRIPTION:Title: The different of a branched cover of $3$-manifolds is a square\nby Mark Shusterman (Harvard) as part of Harvard number theory seminar\
n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nHecke has
shown that the different ideal of a number field is a square in the class
group. In joint work with Will Sawin we obtain an analogous result for clo
sed $3$-manifolds saying that the branch divisor of a covering is a square
in the first homology group.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke)
DTSTART:20230322T190000Z
DTEND:20230322T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/78
DESCRIPTION:Title: A polynomial sieve: beyond separation of variables\nby Lilli
an Pierce (Duke) as part of Harvard number theory seminar\n\nLecture held
in Room 507 in the Science Center.\n\nAbstract\nMany problems in number th
eory can be framed as questions about counting solutions to a Diophantine
equation (say\, within a certain “box”). If there are very few\, or ve
ry many variables\, certain methods gain an advantage\, but sometimes ther
e is extra structure that can be exploited as well. For example: let $f$ b
e a given polynomial with integer coefficients in $n$ variables. How many
values of $f$ are a perfect square? A perfect cube? Or\, more generally\,
a value of a different polynomial of interest\, say $g(y)$? These question
s arise in a variety of specific applications\, and also in the context of
a general conjecture of Serre on counting points in thin sets. We will de
scribe how sieve methods can exploit this type of structure\, and explain
how a new polynomial sieve method allows greater flexibility\, so that the
variables in the polynomials $f$ and $g$ can “mix.” This is joint wor
k with Dante Bonolis.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Ho (Princeton / IAS)
DTSTART:20230329T190000Z
DTEND:20230329T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/79
DESCRIPTION:Title: Selmer averages in families of elliptic curves and applications<
/a>\nby Wei Ho (Princeton / IAS) as part of Harvard number theory seminar\
n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nOrbits of
many coregular representations of algebraic groups are closely linked to m
oduli spaces of genus one curves with extra data. We may use these orbit p
arametrizations to compute the average size of Selmer groups of elliptic c
urves in certain families\, e.g.\, with marked points\, thus obtaining upp
er bounds for the average ranks of the elliptic curves in these families.
(This is joint work with Manjul Bhargava.) We will also describe some othe
r applications and related work (some joint with collaborators\, including
Levent Alpöge\, Manjul Bhargava\, Tom Fisher\, Jennifer Park).\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Newton (King's College London)
DTSTART:20230405T190000Z
DTEND:20230405T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/80
DESCRIPTION:Title: Evaluating the wild Brauer group\nby Rachel Newton (King's C
ollege London) as part of Harvard number theory seminar\n\nLecture held in
Room 507 in the Science Center.\n\nAbstract\nThe local-global approach to
the study of rational points on varieties over number fields begins by em
bedding the set of rational points on a variety $X$ into the set of its ad
elic points. The Brauer–Manin pairing cuts out a subset of the adelic po
ints\, called the Brauer–Manin set\, that contains the rational points.
If the set of adelic points is non-empty but the Brauer–Manin set is emp
ty then we say there's a Brauer–Manin obstruction to the existence of ra
tional points on $X$. Computing the Brauer–Manin pairing involves evalua
ting elements of the Brauer group of $X$ at local points. If an element of
the Brauer group has order coprime to $p$\, then its evaluation at a $p$-
adic point factors via reduction of the point modulo $p$. For elements of
order a power of $p$\, this is no longer the case: in order to compute the
evaluation map one must know the point to a higher $p$-adic precision. Cl
assifying Brauer group elements according to the precision required to eva
luate them at $p$-adic points gives a filtration which we describe using w
ork of Kato. Applications of our work include addressing Swinnerton-Dyer's
question about which places can play a role in the Brauer–Manin obstruc
tion. This is joint work with Martin Bright.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Harris (Columbia)
DTSTART:20230412T190000Z
DTEND:20230412T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/81
DESCRIPTION:Title: Square root $p$-adic $L$-functions\nby Michael Harris (Colum
bia) as part of Harvard number theory seminar\n\nLecture held in Room 507
in the Science Center.\n\nAbstract\nThe Ichino–Ikeda conjecture\, and it
s generalization to unitary groups by N. Harris\, gives explicit formulas
for central critical values of a large class of Rankin–Selberg tensor pr
oducts. The version for unitary groups is now a theorem\, and expresses th
e central critical value of $L$-functions of the form $L(s\,\\Pi \\times \
\Pi')$ in terms of squares of automorphic periods on unitary groups. Here
$\\Pi \\times \\Pi'$ is an automorphic representation of $\\mathrm{GL}(n\
,F)\\times\\mathrm{GL}(n-1\,F)$ that descends to an automorphic representa
tion of $\\mathrm{U}(V) \\times \\mathrm{U}(V')$\, where $V$ and $V'$ are
hermitian spaces over $F$\, with respect to a Galois involution $c$ of $F$
\, of dimension $n$ and $n-1$\, respectively.\n\nI will report on the cons
truction of a $p$-adic interpolation of the automorphic period — in othe
r words\, of the square root of the central values of the $L$-functions
— when $\\Pi'$ varies in a Hida family. The construction is based on the
theory of $p$-adic differential operators due to Eischen\, Fintzen\, Mant
ovan\, and Varma. Most aspects of the construction should generalize to hi
gher Hida theory. I will explain the archimedean theory of the expected ge
neralization\, which is the subject of work in progress with Speh and Koba
yashi.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi (Boston College)
DTSTART:20230419T190000Z
DTEND:20230419T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/82
DESCRIPTION:Title: Derived cycles on Shimura varieties\nby Keerthi Madapusi (Bo
ston College) as part of Harvard number theory seminar\n\nLecture held in
Room 507 in the Science Center.\n\nAbstract\nI will show how methods from
derived algebraic geometry can be used to give a uniform definition of gen
erating series of cycles on integral models of Shimura varieties of Hodge
or even abelian type. Following conjectures of Kudla\, these series are ex
pected to converge to half-integer weight automorphic forms on split unita
ry groups\, and certain ‘easy’ consequences of this expectation turn o
ut to be indeed easy given the derived perspective.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomer Schlank (Hebrew University of Jerusalem)
DTSTART:20230426T190000Z
DTEND:20230426T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/83
DESCRIPTION:Title: Knots Invariants and Arithmetic Statistics\nby Tomer Schlank
(Hebrew University of Jerusalem) as part of Harvard number theory seminar
\n\nLecture held in Room 507 in the Science Center.\n\nAbstract\nThe Groth
endieck school introduced étale topology to attach algebraic-topological
invariants such as cohomology to varieties and schemes. Although the origi
nal motivations came from studying varieties over fields\, interesting phe
nomena such as Artin–Verdier duality also arise when considering the spe
ctra of integer rings in number fields and related schemes. A deep insight
\, due to B. Mazur\, is that through the lens of étale topology\, spectra
of integer rings behave as $3$-dimensional manifolds while prime ideals c
orrespond to knots in these manifolds. This knots and primes analogy provi
des a dictionary between knot theory and number theory\, giving some surpr
ising analogies. For example\, this theory relates the linking number to t
he Legendre symbol and the Alexander polynomial to Iwasawa theory. In thi
s talk\, we shall start by describing some of the classical ideas in this
theory. I shall then proceed by describing how via this theory\, giving a
random model on knots and links can be used to predict the statistical beh
avior of arithmetic functions. This is joint work with Ariel Davis.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjorn Poonen (MIT)
DTSTART:20231018T190000Z
DTEND:20231018T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/84
DESCRIPTION:Title: Integral points on curves via Baker's method and finite étale c
overs\nby Bjorn Poonen (MIT) as part of Harvard number theory seminar\
n\nLecture held in Science Center Room 507.\n\nAbstract\nWe prove results
in the direction of showing that for some affine\ncurves\, Baker's method
applied to finite étale covers is insufficient to\ndetermine the integral
points. This is joint work with Aaron Landesman.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zureick-Brown (Amherst College)
DTSTART:20231108T200000Z
DTEND:20231108T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/85
DESCRIPTION:Title: $\\ell$-adic images of Galois for elliptic curves over $\\mathbb
{Q}$\nby David Zureick-Brown (Amherst College) as part of Harvard numb
er theory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\
nI will discuss recent joint work with Jeremy Rouse and Drew Sutherland on
Mazur’s “Program B” — the classification of the possible “image
s of Galois” associated to an elliptic curve (equivalently\, classificat
ion of all rational points on certain modular curves $X_H$). The main resu
lt is a provisional classification of the possible images of $\\ell$-adic
Galois representations associated to elliptic curves over $\\mathbb{Q}$ an
d is provably complete barring the existence of unexpected rational points
on modular curves associated to the normalizers of non-split Cartan subgr
oups and two additional genus 9 modular curves of level 49.\n\nI will also
discuss the framework and various applications (for example: a very fast
algorithm to rigorously compute the $\\ell$-adic image of Galois of an ell
iptic curve over $\\mathbb{Q}$)\, and then highlight several new ideas fro
m the joint work\, including techniques for computing models of modular cu
rves and novel arguments to determine their rational points\, a computatio
nal approach that works directly with moduli and bypasses defining equatio
ns\, and (with John Voight) a generalization of Kolyvagin’s theorem to t
he modular curves we study.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Drew Sutherland (MIT)
DTSTART:20231206T200000Z
DTEND:20231206T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/86
DESCRIPTION:Title: L-functions from nothing\nby Drew Sutherland (MIT) as part o
f Harvard number theory seminar\n\nLecture held in Science Center Room 507
.\n\nAbstract\nI will report on joint work in progress with Andrew Booker
on\nthe practical implementation of an axiomatic approach to the enumerati
on\nof arithmetic $L$-functions that lie in a certain subset of the Selber
g\nclass that is expected to include all $L$-functions of abelian varietie
s.\nAs in the work of Farmer\, Koutsoliotas\, and Lemurell\, our approach
is\nbased on the approximate functional equation. We obtain additional\nc
onstraints by considering twists (and more general Rankin-Selberg\nconvolu
tions) of our unknown $L$-function that yield a system of linear\nconstrai
nts that can be solved using the simplex method. This allows us\nto signi
ficantly extend the range of our computations for the family of\n$L$-funct
ions associated to abelian surfaces over $\\mathbb{Q}$. We also introduce
a\nmethod for certifying the completeness of our enumeration.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (ICERM)
DTSTART:20231025T190000Z
DTEND:20231025T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/87
DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieti
es\nby Padmavathi Srinivasan (ICERM) as part of Harvard number theory
seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\n$p$-adic
heights have been a rich source of explicit functions vanishing on rationa
l points on a curve. In this talk\, we will outline a new construction of
canonical $p$-adic heights on abelian varieties from $p$-adic adelic metri
cs\, using $p$-adic Arakelov theory developed by Besser. This construction
closely mirrors Zhang's construction of canonical real valued heights fro
m real-valued adelic metrics. We will use this new construction to give di
rect explanations (avoiding $p$-adic Hodge theory) of the key properties o
f height pairings needed for the quadratic Chabauty method for rational po
ints. This is joint work with Amnon Besser and Steffen Mueller.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ananth Shankar (Northwestern University)
DTSTART:20231115T200000Z
DTEND:20231115T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/88
DESCRIPTION:Title: Semisimplicity and CM lifts\nby Ananth Shankar (Northwestern
University) as part of Harvard number theory seminar\n\nLecture held in S
cience Center Room 507.\n\nAbstract\nConsider the setting of a smooth vari
ety $S$ over $\\mathbb{F}_q$\, and an $\\ell$-adic local on $S$ which has
finite determinant and is geometrically irreducible. Work of Lafforgue pro
ves that such a local system must be pure\, and it is conjectured that the
action of Frobenius at closed points is semisimple. I will sketch a proof
of this conjecture in the setting of mod $p$ Shimura varieties\, and will
deduce applications to the existence of CM lifts of certain mod p points.
If time permits\, I will also address the question of integral canonical
models of Shimura varieties.\nThis is joint work with Ben Bakker and Jacob
Tsimerman.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Weston (UMass Amherst)
DTSTART:20230927T190000Z
DTEND:20230927T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/89
DESCRIPTION:Title: Diophantine Stability for Elliptic Curves\nby Tom Weston (UM
ass Amherst) as part of Harvard number theory seminar\n\nLecture held in S
cience Center Room 507.\n\nAbstract\nWe prove\, for any prime $l$ greater
than or equal to 5\, that a density one set of rational elliptic curves ar
e $l$-Diophantine stable in the sense of Mazur and Rubin. This is joint w
ork with Anwesh Ray.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jun Yang (Harvard University)
DTSTART:20231101T190000Z
DTEND:20231101T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/90
DESCRIPTION:Title: The limit multiplicities and von Neumann dimensions\nby Jun
Yang (Harvard University) as part of Harvard number theory seminar\n\nLect
ure held in Science Center Room 507.\n\nAbstract\nGiven an arithmetic subg
roup $\\Gamma$ in a semi-simple Lie group $G$\, the multiplicity of an irr
educible representation of $G$ in $L^2(\\Gamma\\backslash G)$ is unknown i
n general.\nWe observe the multiplicity of any discrete series representat
ion $\\pi$ of $\\rm{SL}(2\,\\mathbb{R})$ in $L^2(\\Gamma(n)\\backslash \\r
m{SL}(2\,\\mathbb{R}))$ is close to the von Neumann dimension of $\\pi$ ov
er the group algebra of $\\Gamma(n)$.\nWe extend this result to other Lie
groups and bounded families of irreducible representations of them.\nBy ap
plying the trace formulas\, we show the multiplicities are exactly the von
Neumann dimensions if we take certain towers of descending lattices in so
me Lie groups.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (ICERM)
DTSTART:20230913T190000Z
DTEND:20230913T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/91
DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieti
es\nby Padmavathi Srinivasan (ICERM) as part of Harvard number theory
seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\n$p$-adic
heights have been a rich source of explicit functions vanishing on rationa
l points on a curve. In this talk\, we will outline a new construction of
canonical $p$-adic heights on abelian varieties from $p$-adic adelic metri
cs\, using $p$-adic Arakelov theory developed by Besser. This construction
closely mirrors Zhang's construction of canonical real valued heights fro
m real-valued adelic metrics. We will use this new construction to give di
rect explanations (avoiding $p$-adic Hodge theory) of the key properties o
f height pairings needed for the quadratic Chabauty method for rational po
ints. This is joint work with Amnon Besser and Steffen Mueller.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Lemke Oliver (Tufts University)
DTSTART:20231129T200000Z
DTEND:20231129T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/92
DESCRIPTION:Title: Faithful induction theorems and the Chebotarev density theorem\nby Robert Lemke Oliver (Tufts University) as part of Harvard number th
eory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nThe
Chebotarev density theorem is a powerful and ubiquitous tool in number the
ory used to guarantee the existence of infinitely many primes satisfying s
plitting conditions in a Galois extension of number fields. In many appli
cations\, however\, it is necessary to know not just that there are many s
uch primes in the limit\, but to know that there are many such primes up t
o a given finite point. This is the domain of so-called effective Chebota
rev density theorems. In forthcoming joint work with Alex Smith that exte
nds previous joint work of the author with Thorner and Zaman and earlier w
ork of Pierce\, Turnage-Butterbaugh\, and Wood\, we prove that in any fami
ly of irreducible complex Artin representations\, almost all are subject t
o a very strong effective prime number theorem. This implies that almost
all number fields with a fixed Galois group are subject to a similarly str
ong effective form of the Chebotarev density theorem. Under the hood\, th
e key result is a new theorem in the character theory of finite groups tha
t is similar in spirit to classical work of Artin and Brauer on inductions
of one-dimensional characters.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Zhang (MIT)
DTSTART:20230920T190000Z
DTEND:20230920T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/93
DESCRIPTION:Title: Harris–Venkatesh plus Stark\nby Robin Zhang (MIT) as part
of Harvard number theory seminar\n\nLecture held in Science Center Room 50
7.\n\nAbstract\nThe class number formula describes the behavior of the Ded
ekind zeta function at $s=0$ and $s=1$. The Stark conjecture extends the c
lass number formula\, describing the behavior of Artin $L$-functions and $
p$-adic $L$-functions at $s=0$ and $s=1$ in terms of units. The Harris–V
enkatesh conjecture describes the residue of Stark units modulo $p$\, givi
ng a modular analogue to the Stark and Gross conjectures while also servin
g as the first verifiable part of the broader conjectures of Venkatesh\, P
rasanna\, and Galatius. In this talk\, I will draw an introductory picture
\, formulate a unified conjecture combining Harris–Venkatesh and Stark f
or weight one modular forms\, and describe the proof of this in the imagin
ary dihedral case.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Betts (Harvard)
DTSTART:20231011T190000Z
DTEND:20231011T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/94
DESCRIPTION:Title: A relative Oda's criterion\nby Alex Betts (Harvard) as part
of Harvard number theory seminar\n\nLecture held in Science Center Hall A.
\n\nAbstract\nThe Neron--Ogg--Shafarevich criterion asserts that an abelia
n variety over $\\mathbb{Q}_p$ has good reduction if and only if the Galoi
s action on its $\\mathbb{Z}_\\ell$-linear Tate module is unramified (for
$\\ell$ different from $p$). In 1995\, Oda formulated and proved an analog
ue of the Neron--Ogg--Shafarevich criterion for smooth projective curves $
X$ of genus at least two: $X$ has good reduction if and only if the outer
Galois action on its pro-$\\ell$ geometric fundamental group is unramified
. In this talk\, I will explain a relative version of Oda's criterion\, du
e to myself and Netan Dogra\, in which we answer the question of when the
Galois action on the pro-$\\ell$ torsor of paths between two points $x$ an
d $y$ is unramified in terms of the relative position of $x$ and $y$ on th
e reduction of $X$. On the way\, we will touch on topics from mapping clas
s groups and the theory of electrical circuits\, and\, time permitting\, w
ill outline some consequences for the Chabauty--Kim method.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART:20231004T190000Z
DTEND:20231004T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/95
DESCRIPTION:Title: Tate Classes and Endoscopy for $\\operatorname{GSp}_4$\nby N
aomi Sweeting (Harvard) as part of Harvard number theory seminar\n\nLectur
e held in Science Center Room 507.\n\nAbstract\nWeissauer proved using the
theory of endoscopy that the Galois representations associated to classic
al modular forms of weight two appear in the middle cohomology of both a m
odular curve and a Siegel modular threefold. Correspondingly\, there are
large families of Tate classes on the product of these two Shimura varieti
es\, and it is natural to ask whether one can construct algebraic cycles g
iving rise to these Tate classes. It turns out that a natural algebraic cy
cle generates some\, but not all\, of the Tate classes: to be precise\, it
generates exactly the Tate classes which are associated to generic member
s of the endoscopic $L$-packets on $\\operatorname{GSp}_4$. In the non-gen
eric case\, one can at least show that all the Tate classes arise from Hod
ge cycles. For this talk\, I'll focus on the behavior of the algebraic cyc
le class. NB: This talk is independent of the one in last week's number th
eorists' seminar.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin-Madison)
DTSTART:20240207T200000Z
DTEND:20240207T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/96
DESCRIPTION:Title: Variation of Selmer groups in quadratic twist families of abelia
n varieties over function fields\nby Jordan Ellenberg (University of W
isconsin-Madison) as part of Harvard number theory seminar\n\nLecture held
in Science Center Room 507.\n\nAbstract\nA basic question in arithmetic s
tatistics is: what does the Selmer group of a random abelian variety look
like? This question is governed by the Poonen-Rains heuristics\, later g
eneralized by Bhargava-Kane-Lenstra-Poonen-Rains\, which predict\, for ins
tance\, that the mod p Selmer group of an elliptic curve has size p+1 on a
verage. Results towards these heuristics have been very partial but have
nonetheless enabled major progress in studying the distribution of ranks o
f abelian varieties.\n\n \n\nWe will describe new work\, joint with Aaron
Landesman\, which establishes a version of these heuristics for the mod n
Selmer group of a random quadratic twist of a fixed abelian variety over a
global function field. This allows us\, for instance\, to bound the prob
ability that a random quadratic twist of an abelian variety A over a globa
l function field has rank at least 2. The method is very much in the spir
it of earlier work with Venkatesh and Westerland which proved a version of
the Cohen-Lenstra heuristics over function fields by means of homological
stabilization for Hurwitz spaces\; in other words\, the main argument is
topological in nature. I will try to embed the talk in a general discussi
on of how one gets from topological results to consequences in arithmetic
statistics\, and what the prospects for further developments in this area
look like.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART:20231122T200000Z
DTEND:20231122T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/97
DESCRIPTION:by TBA as part of Harvard number theory seminar\n\nLecture hel
d in Science Center Room 507.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART:20240424T190000Z
DTEND:20240424T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/98
DESCRIPTION:Title: Shadow line distributions\nby Jennifer Balakrishnan (Boston
University) as part of Harvard number theory seminar\n\nLecture held in Sc
ience Center Room 507.\n\nAbstract\nLet $E/\\mathbb{Q}$ be an elliptic cur
ve of analytic rank $2$\, and let $p$\nbe an odd prime of good\, ordinary
reduction such that the $p$-torsion of\n$E(\\mathbb{Q})$ is trivial. Let $
K$ be an imaginary quadratic field satisfying the\nHeegner hypothesis for
$E$ and such that the analytic rank of the\ntwisted curve $E^K/\\mathbb{Q}
$ is $1$. Further suppose that $p$ splits in $\\mathcal{O}_K$. Under\nthes
e assumptions\, there is a $1$-dimensional $\\mathbb{Q}_p$-vector space at
tached\nto the triple $(E\, p\, K)$\, known as the shadow line\, and it ca
n be\ncomputed using anticyclotomic $p$-adic heights. We describe the\ncom
putation of these heights and shadow lines. Furthermore\, fixing\npairs $
(E\, p)$ and varying $K$\, we present some data on the distribution\nof th
ese shadow lines. This is joint work with Mirela Çiperiani\,\nBarry Mazu
r\, and Karl Rubin.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuan Liu (University of Illinois Urbana-Champaign)
DTSTART:20240417T190000Z
DTEND:20240417T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/99
DESCRIPTION:Title: On the distribution of class groups — beyond Cohen-Lenstra and
Gerth\nby Yuan Liu (University of Illinois Urbana-Champaign) as part
of Harvard number theory seminar\n\nLecture held in Science Center Room 50
7.\n\nAbstract\nThe Cohen-Lenstra heuristic studies the distribution of th
e p-part of the class group of quadratic number fields for odd prime $p$.
Gerth’s conjecture regards the distribution of the $2$-part of the class
group of quadratic fields. The main difference between these conjectures
is that while the (odd) $p$-part of the class group behaves completely “
randomly”\, the $2$-part of the class group does not since the $2$-torsi
on of the class group is controlled by the genus field. In this talk\, we
will discuss a new conjecture generalizing Cohen-Lenstra and Gerth’s con
jectures. The techniques involve Galois cohomology and the embedding probl
em of global fields.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Patrikis (The Ohio State University)
DTSTART:20240214T200000Z
DTEND:20240214T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/101
DESCRIPTION:Title: Compatibility of the canonical $l$-adic local systems on except
ional Shimura varieties\nby Stefan Patrikis (The Ohio State University
) as part of Harvard number theory seminar\n\nLecture held in Science Cent
er Room 507.\n\nAbstract\nLet $(G\, X)$ be a Shimura datum\, and let $K$ b
e a compact open subgroup of $G(\\mathbb{A}_f)$. One hopes that under mild
assumptions on $G$ and $K$\, the points of the Shimura variety $Sh_K(G\,
X)$ parametrize a family of motives\; in abelian type this is well-underst
ood\, but in non-abelian type it is completely mysterious. I will discuss
joint work with Christian Klevdal showing that for exceptional Shimura var
ieties the points (over number fields\, say) at least yield compatible sys
tems of l-adic representations\, which should be the l-adic realizations o
f the conjectural motives.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (MIT)
DTSTART:20240221T200000Z
DTEND:20240221T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/102
DESCRIPTION:Title: The Average Size of 2-Selmer Groups of Elliptic Curves over Fun
ction Fields\nby Niven Achenjang (MIT) as part of Harvard number theor
y seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nGiven a
n elliptic curve $E$ over a global field $K$\, the abelian group $E(K)$ is
finitely generated\, and so much effort has been put into trying to under
stand the behavior of $\\operatorname{rank}E(K)$\, as $E$ varies. Of note\
, it is a folklore conjecture that\, when all elliptic curves $E/K$ are or
dered by a suitably defined height\, the average value of their ranks is e
xactly $1/2$. One fruitful avenue for understanding the distribution of $\
\operatorname{rank}E(K)$ has been to first understand the distribution of
the sizes of Selmer groups of elliptic curves. In this direction\, various
authors (including Bhargava-Shankar\, Poonen-Rains\, and Bhargava-Kane-Le
nstra-Poonen-Rains) have made conjectures which predict\, for example\, th
at the average size of the $n$-Selmer group of $E/K$ is equal to the sum o
f the divisors of $n$. In this talk\, I will report on some recent work ve
rifying this average size prediction\, "up to small error term\," whenever
$n=2$ and $K$ is any global *function* field. Results along these lines w
ere previously known whenever $K$ was a number field or function field of
characteristic $\\ge 5$\, so the novelty of my work is that it applies eve
n in "bad" characteristic.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Mundy (Princeton University)
DTSTART:20240410T190000Z
DTEND:20240410T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/103
DESCRIPTION:Title: Vanishing of Selmer groups for Siegel modular forms\nby Sam
Mundy (Princeton University) as part of Harvard number theory seminar\n\n
Lecture held in Science Center Room 507.\n\nAbstract\nLet $\\pi$ be a cusp
idal automorphic representation of $\\mathrm{Sp}_{2n}$ over $\\mathbb{Q}$
which is holomorphic discrete series at infinity\, and $\\chi$ a Dirichlet
character. Then one can attach to $\\pi$ an orthogonal $p$-adic Galois re
presentation $\\rho$ of dimension $2n+1$. Assume $\\rho$ is irreducible\,
that $\\pi$ is ordinary at $p$\, and that $p$ does not divide the conducto
r of $\\chi$. I will describe work in progress which aims to prove that th
e Bloch--Kato Selmer group attached to the twist of $\\rho$ by $\\chi$ van
ishes\, under some mild ramification assumptions on $\\pi$\; this is what
is predicted by the Bloch--Kato conjectures.\n\n\nThe proof uses "ramified
Eisenstein congruences" by constructing $p$-adic families of Siegel cusp
forms degenerating to Klingen Eisenstein series of nonclassical weight\, a
nd using these families to construct ramified Galois cohomology classes fo
r the Tate dual of the twist of $\\rho$ by $\\chi$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART:20240228T200000Z
DTEND:20240228T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/104
DESCRIPTION:Title: Computing Galois images of Picard curves\nby Shiva Chidamba
ram (MIT) as part of Harvard number theory seminar\n\nLecture held in Scie
nce Center Room 507.\n\nAbstract\nLet $C$ be a genus $3$ curve whose Jacob
ian is geometrically simple and has geometric endomorphism algebra equal t
o an imaginary quadratic field. In particular\, consider Picard curves $y^
3 = f_4(x)$ where the geometric endomorphism algebra is $\\mathbb{Q}(\\zet
a_3)$. We study the associated mod-$\\ell$ Galois representations and thei
r images. I will discuss an algorithm\, developed in ongoing joint work wi
th Pip Goodman\, to compute the set of primes $\\ell$ for which the images
are not maximal. By running it on several datasets of Picard curves\, the
largest non-maximal prime we obtain is $13$. This may be compared with ge
nus 1\, where Serre's uniformity question asks if the mod-$\\ell$ Galois i
mage of non-CM elliptic curves over $\\Q$ is maximal for all primes $\\ell
> 37$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (Harvard University)
DTSTART:20240501T190000Z
DTEND:20240501T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/106
DESCRIPTION:Title: Modularity of special cycles in orthogonal and unitary Shimura
varieties\nby Salim Tayou (Harvard University) as part of Harvard numb
er theory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\
nSince the work of Jacobi and Siegel\, it is well known that\nTheta series
of quadratic lattices produce modular forms. In a vast\ngeneralization\,
Kudla and Millson have proved that the generating series\nof special cycle
s in orthogonal and unitary Shimura varieties are\nmodular forms. In this
talk\, I will explain an extension of these\nresults to toroidal compactif
ications where we prove that the generating\nseries of divisors is a mixed
mock modular form. This recovers and\nrefines earlier results of Bruinier
and Zemel. The results of this talk\nare joint work with Philip Engel and
François Greer.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART:20240327T190000Z
DTEND:20240327T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/107
DESCRIPTION:Title: “everywhere unramified” objects in number theory and the co
homology of $\\mathrm{GL}_n(\\mathbb{Z})$\nby Frank Calegari (Universi
ty of Chicago) as part of Harvard number theory seminar\n\nLecture held in
Science Center Room 507.\n\nAbstract\nOne theme in number theory is to st
udy objects via their ramification: the discriminant of a number field\, t
he conductor of an elliptic curve\, the level of a modular form\, and so o
n.\nThere is\, however\, some particular interest in understanding objects
which are “everywhere unramified” — and also understanding when suc
h objects don’t exist. Such non-existence results\nare often the startin
g point for inductive arguments. For example\, Minkowski’s theorem that
there are no unramified extensions of $\\mathbb{Q}$ can be used to prove t
he Kronecker-Weber theorem\, and the vanishing\nof a certain space of modu
lar forms is the starting point for Wiles’ proof of Fermat’s Last Theo
rem. In this talk\, I will begin by describing many such vanishing results
both in arithmetic and in the\ntheory of automorphic forms\, and how they
are related by the Langlands program (sometimes only conjecturally). Then
I will descibe the construction of a new example of an automorphic form o
f level one\nand “weight zero”. This construction also gives the firs
t non-zero classes in the cohomology of $\\mathrm{GL}_n(\\mathbb{Z})$ (for
some $n$) that come from “cuspidal” modular forms (for $n > 0$).\n\nT
his is joint work with George Boxer and Toby Gee.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART:20240306T200000Z
DTEND:20240306T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/109
DESCRIPTION:Title: Integral Ax-Sen-Tate theory\nby Jared Weinstein (Boston Uni
versity) as part of Harvard number theory seminar\n\nLecture held in Scien
ce Center Room 507.\n\nAbstract\nLet $K$ be a local field of mixed charact
eristic\, let $G$ be the absolute Galois group of $K$\, and let $C$ be the
completion of an algebraic closure of $K$. The Ax-Sen-Tate theorem state
s that the field of $G$-invariant elements in $C$ is $K$ itself: $H^0(G\,
C)=K$. Tate also proved statements about higher cohomology (with continuo
us cocycles): $H^1(G\,C)=K$ and $H^i(G\,C)=0$ for $i>1$. \n Let $O_C$
be the ring of integers in $C$. Our main theorem is that the torsion sub
group of $H^i(G\,O_C)$ is killed by a constant which only depends on the r
esidue characteristic $p$ (in fact $p^6$ suffices). This is a part of a p
roject with coauthors Tobias Barthel\, Tomer Schlank\, and Nathaniel Stapl
eton.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sasha Petrov (MIT)
DTSTART:20240911T190000Z
DTEND:20240911T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/110
DESCRIPTION:Title: Characteristic classes of p-adic local systems\nby Sasha Pe
trov (MIT) as part of Harvard number theory seminar\n\nLecture held in Sci
ence Center Room 507.\n\nAbstract\nGiven an étale Z_p-local system of ran
k n on an algebraic variety X\, continuous cohomology classes of the group
GL_n(Z_p) give rise to classes in (absolute) étale cohomology of the var
iety with coefficients in Q_p. These characteristic classes can be thought
of as p-adic analogs of Chern-Simons characteristic classes of vector bun
dles with a flat connection.\n\nOn a smooth projective variety over comple
x numbers\, Chern-Simons classes of all flat bundles are torsion in degree
s >1 by a theorem of Reznikov. But for varieties over non-closed fields th
e characteristic classes of p-adic local systems turn out to often be non-
zero even rationally. When X is defined over a p-adic field\, characterist
ic classes of a p-adic local system on it can be partially expressed in te
rms of Hodge-theoretic invariants of the local system. This relation is es
tablished through considering an analog of Chern classes for vector bundle
s on the pro-étale site of X.\n\nThis is joint work with Lue Pan.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jit Wu Yap (Harvard University)
DTSTART:20240918T190000Z
DTEND:20240918T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/111
DESCRIPTION:Title: Quantitative Equidistribution of Small Points for Canonical Hei
ghts\nby Jit Wu Yap (Harvard University) as part of Harvard number the
ory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nLet K
be a number field with algebraic closure L and A an abelian variety over
K. Then if (x_n) is a generic sequence of points of A(L) with Neron-Tate h
eight tending to 0\, Szpiro-Ullmo-Zhang proved that the Galois orbits of x
_n converges weakly to the Haar measure of A. Yuan then generalized Szpiro
-Ullmo-Zhang's result to the setting of polarized endomorphisms on a proje
ctive variety X defined over K. In this talk\, I will explain how to prove
a quantitative version of Yuan's result when X is assumed to be smooth. T
his was previously only known when dim X = 1.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (Freie Universität Berlin)
DTSTART:20240925T190000Z
DTEND:20240925T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/112
DESCRIPTION:Title: Diophantine Properties of the Betti Moduli Space\nby Hélè
ne Esnault (Freie Universität Berlin) as part of Harvard number theory se
minar\n\nLecture held in Science Center Room 507.\n\nAbstract\nWe prove in
particular that when the Betti moduli space of a smooth quasi-projective
variety\nover the complex number with some quasi-unipotent monodromies at
infinity. finite determinant\nis irreducible over the integers and over th
e complex numbers\, then it possesses an integral point. \nA more general
version of the theorem yields a new obstruction for the finitely presented
group to be the topological fundamental group\nof a smooth complex quasi-
projective variety. \n\n(Joint with J. de Jong\, based in part on joint wo
rk with M. Groechenig).\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanath Devalapurkar (Harvard University)
DTSTART:20241002T190000Z
DTEND:20241002T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/113
DESCRIPTION:Title: The image of J and p-adic geometry\nby Sanath Devalapurkar
(Harvard University) as part of Harvard number theory seminar\n\nLecture h
eld in Science Center Room 507.\n\nAbstract\nFor a prime p\, Bhatt\, Lurie
\, and Drinfeld constructed the "prismatization" of a p-adic formal scheme
\; this is a stack which computes prismatic cohomology\, which is a "unive
rsal" cohomology theory for p-adic formal schemes. I will describe joint w
ork with Hahn\, Raksit\, and Yuan (building on work of Hahn-Raksit-Wilson)
\, in which we give a new construction of prismatization using the methods
of homotopy theory (in particular\, the theory of topological Hochschild
homology\, aka THH). The case when R is Z_{p} turns out to be particularly
interesting\, and I will discuss joint work with Raksit which describes a
construction of THH(Z_{p}) for odd primes p in terms of a very classical
object in homotopy theory called the "image-of-J spectrum" studied by Adam
s. This plays the same role for prismatic cohomology as the usual commutat
ive ring Z_{p} plays for crystalline cohomology. It gives an alternative p
erspective on results of Bhatt and Lurie\, and is also related to Lurie’
s "prismatization of F_{1}".\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sameera Vemulapalli (Harvard University)
DTSTART:20241009T190000Z
DTEND:20241009T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/114
DESCRIPTION:Title: Steinitz classes of number fields and Tschirnhausen bundles of
covers of the projective line\nby Sameera Vemulapalli (Harvard Univers
ity) as part of Harvard number theory seminar\n\nLecture held in Science C
enter Room 507.\n\nAbstract\nGiven a number field extension $L/K$ of fixed
degree\, one may consider $\\mathcal{O}_L$ as an $\\mathcal{O}_K$-module.
Which modules arise this way? Analogously\, in the geometric setting\, a
cover of the complex projective line by a smooth curve yields a vector bun
dle on the projective line by pushforward of the structure sheaf\; which b
undles arise this way? In this talk\, I'll describe recent work with Vakil
in which we use tools in arithmetic statistics (in particular\, binary fo
rms) to completely answer the first question and make progress towards the
second.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Artane Siad (Princeton University)
DTSTART:20241016T190000Z
DTEND:20241016T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/115
DESCRIPTION:Title: Spin structures\, quadratic maps\, and the missing class group
heuristic\nby Artane Siad (Princeton University) as part of Harvard nu
mber theory seminar\n\nLecture held in Science Center Room 507.\n\nAbstrac
t\nI will report on joint work in progress with Akshay Venkatesh where we
propose an arithmetic analogue of the association\, in topology\, of quadr
atic enhancements to spin structures on closed oriented 2- and 3-manifolds
: a choice of spin structure provides\, respectively\, a quadratic refinem
ent of the mod 2 intersection form and of the linking pairing on the first
torsion homology. This adds an entry to the number field/3-manifold analo
gy of Mumford\, Mazur\, and Manin and furnishes a conceptual explanation o
f anomalous class group statistics.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Hamann (Harvard University)
DTSTART:20241023T190000Z
DTEND:20241023T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/116
DESCRIPTION:Title: Shimura Varieties and Eigensheaves\nby Linus Hamann (Harvar
d University) as part of Harvard number theory seminar\n\nLecture held in
Science Center Room 507.\n\nAbstract\nThe cohomology of Shimura varieties
is a fundamental object of study in algebraic number theory by virtue of t
he fact that it is the only known geometric realization of the global Lang
lands correspondence over number fields. Usually\, the cohomology is compu
ted through very delicate techniques involving the trace formula. However\
, this perspective has several limitations\, especially with regards to qu
estions concerning torsion. In this talk\, we will discuss a new paradigm
for computing the cohomology of Shimura varieties by decomposing certain
sheaves coming from Igusa varieties into Hecke eigensheaves on the moduli
stack of G-bundles on the Fargues-Fontaine curve. Using this point of view
\, we will describe several conjectures on the torsion cohomology of Shimu
ra varieties after localizing at suitably "generic" L-parameters\, as well
as some known results in the case that the parameter factors through a ma
ximal torus. Motivated by this\, we will sketch part of an emerging pictur
e for describing the cohomology beyond this generic locus by considering c
ertain "generalized eigensheaves" whose eigenvalues are spread out in mult
iple cohomological degrees based on the size of a certain Arthur SL_{2} in
a way that is reminiscent of Arthur's cohomological conjectures on the in
tersection cohomology of Shimura Varieties. This is based on joint work wi
th Lee\, joint work in progress with Caraiani and Zhang\, and conversation
s with Bertoloni-Meli and Koshikawa.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Zhang (MIT)
DTSTART:20241030T190000Z
DTEND:20241030T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/117
DESCRIPTION:Title: Faltings heights and the sub-leading terms of adjoint L-functio
ns\nby Wei Zhang (MIT) as part of Harvard number theory seminar\n\nLec
ture held in Science Center Room 507.\n\nAbstract\nBased on work in progre
ss with Ryan Chen and Weixiao Lu.\nThe Kronecker limit formula may be inte
rpreted as an equality relating the Faltings height of an CM elliptic curv
e to the sub-leading term (at s=0) of the Dirichlet L-function of an imagi
nary quadratic character. Colmez conjectured a generalization relating the
Faltings height of any CM abelian variety to the sub-leading terms of cer
tain Artin L-functions. In this talk we will formulate a “non-Artinian
” generalization of (averaged) Colmez conjecture\, relating the followin
g two quantities:\n\n(1) the Faltings height of certain cycles on unitary
Shimura varieties\, and \n(2) the sub-leading terms of the adjoint L-funct
ions of (cohomological) automorphic representations of unitary groups U(n)
. \n\nThe case $n=1$ amounts to the averaged Colmez conjecture. We formula
te a relative trace formula approach for the general $n$\, and we are able
to prove our conjecture when $n=2$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sachi Hashimoto (Brown University)
DTSTART:20241106T200000Z
DTEND:20241106T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/118
DESCRIPTION:Title: Rational points on $X_0(N)^*$ when $N$ is non-squarefree\nb
y Sachi Hashimoto (Brown University) as part of Harvard number theory semi
nar\n\nLecture held in Science Center Room 507.\n\nAbstract\nThe rational
points of the modular curve $X_0(N)$ classify pairs $(E\,C_N)$ of elliptic
curves over $\\mathbb{Q}$ together with a rational cyclic subgroup of ord
er $N$. The curve $X_0(N)^*$ is the quotient of $X_0(N)$ by the full group
of Atkin-Lehner involutions. Elkies showed that the rational points on th
is curve classify elliptic curves over the algebraic closure of $\\mathbb{
Q}$ that are isogenous to their Galois conjugates. In ongoing joint work w
ith Timo Keller and Samuel Le Fourn\, we study the rational points on the
family $X_0(N)^*$ for $N$ non-squarefree. In particular we will report on
some integrality results for $X_0(N)^*$. Our strategy follows the work of
Mazur\, Momose\, and Bilu-Parent-Rebolledo for the families $X_0(p)$ and $
X_0(p^r)^+$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Bertoloni Meli (Boston University)
DTSTART:20241113T200000Z
DTEND:20241113T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/119
DESCRIPTION:Title: Hecke Eigensheaves for Arthur Parameters\nby Alexander Bert
oloni Meli (Boston University) as part of Harvard number theory seminar\n\
nLecture held in Science Center Room 507.\n\nAbstract\nI will talk about w
ork relating to categorical Langlands for non-archimedean local fields. I
n particular\, I will discuss progress with Teruhisa Koshikawa on defining
the Galois-side incarnation of a Hecke eigensheaf attached to an Arthur p
arameter. We will focus primarily on the PGL2 case where everything can be
understood explicitly.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Zubrilina (Harvard University)
DTSTART:20241120T200000Z
DTEND:20241120T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/120
DESCRIPTION:Title: Root Number Correlation Bias of Fourier Coefficients of Modular
Forms\nby Nina Zubrilina (Harvard University) as part of Harvard numb
er theory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\
nIn a recent study\, He\, Lee\, Oliver\, and Pozdnyakov observed a strikin
g oscillating pattern in the average value of the p-th Frobenius trace of
elliptic curves of prescribed rank and conductor in an interval range. Sut
herland discovered that this bias extends to Dirichlet coefficients of a m
uch broader class of arithmetic L-functions when split by root number. In
my talk\, I will discuss this root number correlation in families of holom
orphic and Maass forms.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Smith (UCLA)
DTSTART:20241204T200000Z
DTEND:20241204T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/121
DESCRIPTION:Title: The distribution of conjugates of an algebraic integer\nby
Alex Smith (UCLA) as part of Harvard number theory seminar\n\nLecture held
in Science Center Room 507.\n\nAbstract\nFor every odd prime p\, the numb
er 2 + 2cos(2 pi/p) is an algebraic integer whose conjugates are all posit
ive numbers\; such a number is known as a totally positive algebraic integ
er. For large p\, the average of the conjugates of this number is close to
2\, which is small for a totally positive algebraic integer. The Schur-Si
egel-Smyth trace problem\, as posed by Borwein in 2002\, is to show that n
o sequence of totally positive algebraic integers could best this bound.\n
\nIn this talk\, we will resolve this problem in an unexpected way by cons
tructing infinitely many totally positive algebraic integers whose conjuga
tes have an average of at most 1.899. To do this\, we will apply a new met
hod for constructing algebraic integers to an example first considered by
Serre. We also will explain how our method can be used to find simple abel
ian varieties with extreme point counts.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Shnidman (IAS)
DTSTART:20250129T200000Z
DTEND:20250129T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/122
DESCRIPTION:Title: Hilbert's 10th problem over number fields\nby Ari Shnidman
(IAS) as part of Harvard number theory seminar\n\nLecture held in Science
Center Room 507.\n\nAbstract\nWe show that for every quadratic extension o
f number fields K/F\, there exists an abelian variety A/F of positive rank
whose rank does not grow upon base change to K. This result is known to i
mply that Hilbert's tenth problem over the ring of integers R of any numbe
r field has a negative solution. That is\, there does not exist an algori
thm that answers the question of whether a polynomial equation in several
variables over R has solutions in R. In the pretalk\, I'll talk about CM a
belian varieties and Selmer groups. This is joint work with Levent Alpöge
\, Manjul Bhargava\, and Wei Ho.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sean Howe (University of Utah)
DTSTART:20250205T200000Z
DTEND:20250205T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/123
DESCRIPTION:Title: Inscription and p-adic twistors\nby Sean Howe (University o
f Utah) as part of Harvard number theory seminar\n\nLecture held in Scienc
e Center Room 507.\n\nAbstract\nInspired by a construction of Simpson for
irreducible local systems over compact Kahler manifolds\, both Fargues and
Liu-Zhu have conjectured that p-adic local systems on smooth rigid analyt
ic varieties over p-adic fields should admit associated p-adic twistor bun
dles. We formulate and prove a version of this conjecture using the theory
of inscribed v-sheaves\, which is a simple differential extension of Scho
lze’s approach to p-adic geometry by replacing a classical object with i
ts functor-of-points on perfectoid spaces. As an application\, we explain
how to obtain a non-trivial inscribed structure on p-adic Lie torsors over
smooth rigid analytic varieties that allows us\, in particular\, to compu
te Banach-Colmez Tangent Bundles and differentiate Hodge-Tate period maps
and their lattice refinements. In the case of infinite level local and glo
bal Shimura varieties this agrees with a natural inscribed structure const
ructed by extending a moduli interpretation to the inscribed setting.\n\n
\n\nPretalk: Modern p-adic geometry\n\nAbstract: The basic building blocks
of p-adic geometry have shifted in the past fifteen years from the Noethe
rian convergent power series rings of Tate’s theory of rigid analytic sp
aces\, which mirrors the classical theory of complex analytic spaces\, to
the more exotic perfectoid rings that provide the test objects in Scholze
’s theory of diamonds and v-sheaves and are characterized by the existen
ce of approximate p-power roots. We will give some simple examples contras
ting the behaviors of these types of rings and discuss some of the reasons
for this shift in perspectives.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ryan Chen (MIT)
DTSTART:20250212T200000Z
DTEND:20250212T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/124
DESCRIPTION:Title: Near center derivatives and arithmetic $1$-cycles\nby Ryan
Chen (MIT) as part of Harvard number theory seminar\n\nLecture held in Sci
ence Center Room 507.\n\nAbstract\nDegrees of arithmetic special cycles on
Shimura varieties are expected to appear in first derivatives of automorp
hic forms and L-functions\, such as in the Gross--Zagier formula\, Kudla's
program\, and the Arithmetic Gan--Gross--Prasad program.\n\nI will explai
n some “near-central” instances of an arithmetic Siegel--Weil formula
from Kudla’s program\, which "geometrize" the classical Siegel mass and
Siegel--Weil formulas\, on lattice and lattice vector counting.\n \nAt the
se near-central points of functional symmetry\, it is typical that both th
e "leading" special value (complex volumes) and the "subleading" first der
ivative (arithmetic volume) simultaneously have geometric meaning.\n\nThe
key input is a new "limit phenomenon" relating positive characteristic int
ersection numbers and heights in mixed characteristic\, as well as its aut
omorphic counterpart.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Princeton University)
DTSTART:20250219T200000Z
DTEND:20250219T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/125
DESCRIPTION:Title: Some cases of the Bloch-Kato conjecture for four-dimensional sy
mplectic Galois representations\nby Naomi Sweeting (Princeton Universi
ty) as part of Harvard number theory seminar\n\nLecture held in Science Ce
nter Room 507.\n\nAbstract\nThe Bloch-Kato conjecture is a far-reaching ge
neralization of the famous conjecture of Birch and Swinnerton-Dyer on L fu
nctions of elliptic curves. This talk is about recent results towards Bloc
h-Kato in rank 0 and 1 for spin L-functions of certain automorphic represe
ntations of $\\operatorname{GSp}_4$. I'll explain the statements and some
ideas of the proof\, which is based on constructing ramified Galois cohomo
logy classes via level-raising congruences.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Hansen (National University of Singapore)
DTSTART:20250226T200000Z
DTEND:20250226T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/126
DESCRIPTION:Title: The stable Bernstein center\nby David Hansen (National Univ
ersity of Singapore) as part of Harvard number theory seminar\n\nLecture h
eld in Science Center Room 507.\n\nAbstract\nThe Bernstein center of a p-a
dic reductive group G is a beautiful and explicit commutative ring which a
cts on "everything" related to the representation theory of G. In recent y
ears\, the idea has emerged that this ring contains a canonical subring -
the stable Bernstein center - which should be intimately related with the
local Langlands correspondence. However\, while it is easy to define the s
table Bernstein center\, it is very difficult to exhibit elements in this
subring. On the other hand\, recent work of Fargues-Scholze defines anothe
r totally canonical subring of the Bernstein center\, whose construction u
ses V. Lafforgue's theory of excursion operators adapted to the Fargues-Fo
ntaine curve. After reviewing these stories\, I'll sketch a proof that the
Fargues-Scholze subring is actually contained in the stable Bernstein cen
ter\, for all G.\nIn the pretalk\, I'll give a more leisurely introduction
to the Bernstein center.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sho Tanimoto (Nagoya University in Japan)
DTSTART:20250305T200000Z
DTEND:20250305T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/127
DESCRIPTION:Title: Homological stability and Manin’s conjecture\nby Sho Tani
moto (Nagoya University in Japan) as part of Harvard number theory seminar
\n\nLecture held in Science Center Room 507.\n\nAbstract\nI present our on
going proofs for a version of Manin’s conjecture over F_q for q large an
d Cohen—Jones—Segal conjecture over C for rational curves on split qua
rtic del Pezzo surfaces. The proofs share a common method which builds upo
n prior work of Das—Tosteson. The main ingredients of this method are (i
) the construction of bar complexes formalizing the inclusion-exclusion pr
inciple and its point counting estimates\, (ii) dimension estimates for sp
aces of rational curves using conic bundle structures\, (iii) estimates of
error terms using arguments of Sawin based on Katz’s results\, and (iv)
a certain virtual height zeta function revealing the compatibility of bar
complexes and Peyre’s constant. Our argument verifies the heuristic app
roach to Manin’s conjecture over global function fields given by Batyrev
and Ellenberg–Venkatesh. This is joint work with Ronno Das\, Brian Lehm
ann\, and Phil Tosteson with a help by Will Sawin.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Schiavone (MIT)
DTSTART:20250312T190000Z
DTEND:20250312T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/128
DESCRIPTION:Title: Reconstructing genus 4 curves and applications\nby Sam Schi
avone (MIT) as part of Harvard number theory seminar\n\nLecture held in Sc
ience Center Room 507.\n\nAbstract\nWe present a method for recovering the
canonical model of a genus 4 curve from its theta constants. We describe
some applications\, such as gluing genus 2 curves\, computing examples of
explicit modularity for abelian varieties with real multiplication\, and c
omputing Jacobians with complex multiplication. As a final example\, we di
scuss work in progress toward explicitly computing an abelian 4-fold of Mu
mford type. Joint work with Thomas Bouchet\, Jeroen Hanselman\, and Andrea
s Pieper.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia University)
DTSTART:20250326T190000Z
DTEND:20250326T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/129
DESCRIPTION:Title: Hilbert 10 via additive combinatorics\nby Carlo Pagano (Con
cordia University) as part of Harvard number theory seminar\n\nLecture hel
d in Science Center Room 507.\n\nAbstract\nIn 1970 Matiyasevich\, building
on earlier work of Davis--Putnam--Robinson\, proved that every enumerable
subset of Z is Diophantine\, thus showing that Hilbert's 10th problem is
undecidable for Z. The problem of extending this result to the ring of int
egers of number fields (and more generally to finitely generated infinite
rings) has attracted significant attention and\, thanks to the efforts of
many mathematicians\, the task has been reduced to the problem of construc
ting\, for certain quadratic extensions of number fields L/K\, an elliptic
curve E/K with rk(E(L))=rk(E(K))>0. \n\nIn this talk I will explain joint
work with Peter Koymans\, where we combine Green--Tao with 2-descent to c
onstruct the desired elliptic curves\, settling Hilbert 10 for every finit
ely generated infinite ring. The background material used to execute 2-des
cent in a quadratic twist will be explored during the pre-talk.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Tsimerman (University of Toronto)
DTSTART:20250402T190000Z
DTEND:20250402T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/130
DESCRIPTION:Title: Geometric Shafarevich Conjecture for Exceptional Shimura Variet
ies\nby Jacob Tsimerman (University of Toronto) as part of Harvard num
ber theory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract
\nThe Shafarevich conjecture is concerned with finiteness results for fami
lies of g-dimensional principally polarized abelian varieties over a base
B. Famously\, Faltings settled the case of B=O_{K\,S}. In the case where B
is a curve over a finite field\, finiteness can never be true as one may
always compose with Frobenius. In this setting\, to get a theorem one must
consider families up to p-power isogenies.\n\nWe formulate an analogous s
tatement for Exceptional Shimura varieties S\, and describe ongoing work t
o prove it. This is joint work with Ben Bakker and Ananth Shankar.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jerson Caro (Boston University)
DTSTART:20250409T190000Z
DTEND:20250409T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/131
DESCRIPTION:Title: Counting and Finding Rational Points on Surfaces\nby Jerson
Caro (Boston University) as part of Harvard number theory seminar\n\nLect
ure held in Science Center Room 507.\n\nAbstract\nA celebrated result of C
oleman gives an explicit version of Chabauty's theorem\, bounding the numb
er of rational points on curves over number fields via the study of zeros
of p-adic analytic functions. While many developments have extended and re
fined this result\, obtaining analogous explicit bounds for higher-dimensi
onal subvarieties of abelian varieties remains a major challenge.\nIn this
talk\, I will sketch the proof of such an explicit bound for surfaces con
tained in abelian varieties — a step toward a higher-dimensional Chabaut
y--Coleman method. This is joint work with Héctor Pastén.\nI will also d
escribe an application of this method to a computational problem: determin
ing an upper bound for the number of unexpected quadratic points on hypere
lliptic curves of genus 3 defined over Q. I will illustrate the method thr
ough an explicit example where this set can be computed. This is joint wor
k with Jennifer Balakrishnan.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART:20250416T190000Z
DTEND:20250416T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/132
DESCRIPTION:Title: Variations on a theme of Faltings\nby Jared Weinstein (Bost
on University) as part of Harvard number theory seminar\n\nLecture held in
Science Center Room 507.\n\nAbstract\nThe complex unit disc is conformall
y equivalent to the upper half-plane. In 2002\, Faltings proved a p-adic
version: the p-adic unit disc is isomorphic to Drinfeld’s upper half-pl
ane\, up to the action of some profinite groups. We report on some work i
n progress concerning a family of Faltings-style isomorphisms\, occurring
entirely in characteristic p. These concern moduli spaces of formal group
s over a local base where the generic and special fibers have specified he
ights. We were motivated to study these spaces by problems in chromatic h
omotopy theory. This is joint work with many people.\n\nThe pre-talk will
give more details about the original Faltings isomorphism.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Harvard/MIT)
DTSTART:20250423T190000Z
DTEND:20250423T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/133
DESCRIPTION:Title: Arithmetic statistics via homological stability\nby Aaron L
andesman (Harvard/MIT) as part of Harvard number theory seminar\n\nLecture
held in Science Center Room 507.\n\nAbstract\nIn my view\, the three main
conjectures in arithmetic statistics are the Cohen-Lenstra conjectures\,
Malle's conjecture\, and the Poonen-Rains conjectures. We will explain the
statements of these three conjectures and how\, in the function field set
ting\, they are related to understanding the homology of certain Hurwitz s
paces. This is partially an advertisement for my topics course at Harvard
next year and is related to joint work with Ishan Levy and work with Jorda
n Ellenberg.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Litt (University of Toronto)
DTSTART:20250430T190000Z
DTEND:20250430T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/134
DESCRIPTION:Title: On the converse to Eisenstein's last theorem\nby Daniel Lit
t (University of Toronto) as part of Harvard number theory seminar\n\nLect
ure held in Science Center Room 507.\n\nAbstract\nI'll explain a conjectur
al characterization of algebraic solutions to (possibly non-linear) algebr
aic differential equations\, in terms of the arithmetic of the coefficient
s of their Taylor expansions\, strengthening the Grothendieck-Katz p-curva
ture conjecture. I'll give some evidence for the conjecture coming from al
gebraic geometry: in joint work with Josh Lam\, we verify the conjecture f
or algebraic differential equations (both linear and non-linear) and initi
al conditions of algebro-geometric origin. In this case the conjecture tur
ns out to be closely related to basic conjectures on algebraic cycles\, mo
tives\, and so on.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Lemke Oliver (Tufts University)
DTSTART:20250507T190000Z
DTEND:20250507T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/135
DESCRIPTION:Title: Enumerating Galois extensions of number fields\nby Robert L
emke Oliver (Tufts University) as part of Harvard number theory seminar\n\
nLecture held in Science Center Room 507.\n\nAbstract\nLet $k$ be a number
field. We provide an asymptotic formula for the number of Galois extensio
ns of $k$ with absolute discriminant bounded by some $X \\geq 1$ as $X \\t
o \\infty$. The key behind this result is a new upper bound on the numbe
r of Galois extensions of $k$ with a given Galois group $G$ and discrimina
nt bounded by $X$\; we show the number of such extensions is $O_{[k:Q]\,G}
(X^{4/\\sqrt{|G|}})$. This improves over the previous best bound $O_{k\,G\
,\\epsilon}(X^{3/8+\\epsilon})$ due to Ellenberg and Venkatesh. In particu
lar\, ours is the first bound for general $G$ with an exponent that decays
as $|G| \\to \\infty$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jit Wu Yap (MIT)
DTSTART:20250917T190000Z
DTEND:20250917T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/136
DESCRIPTION:Title: On Uniform Boundedness of Torsion Points for Abelian Varieties
over Function Fields\nby Jit Wu Yap (MIT) as part of Harvard number th
eory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nLet
K be the function field of a smooth projective curve B over the complex nu
mbers and let g be a positive integer. The uniform boundedness conjecture
predicts that there exists a constant N\, depending only on g and K\, such
that for any g-dimensional abelian variety A over K\, any K-rational tors
ion point of A must have order at most N. In this talk\, we will discuss s
ome recent progress under the assumption that A has semistable reduction o
ver K. This is joint work with Nicole Looper.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Li-Huerta (MIT)
DTSTART:20250924T190000Z
DTEND:20250924T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/137
DESCRIPTION:Title: Close fields and the local Langlands correspondence\nby Dan
iel Li-Huerta (MIT) as part of Harvard number theory seminar\n\nLecture he
ld in Science Center Room 507.\n\nAbstract\nThere is an idea\, going back
to work of Krasner\, that $p$-adic fields tend to function fields as absol
ute ramification tends to infinity. We will present a new way of rigorizin
g this idea\, as well as give applications to the local Langlands correspo
ndence of Fargues–Scholze.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Lin (Harvard University)
DTSTART:20251001T190000Z
DTEND:20251001T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/138
DESCRIPTION:Title: Finiteness of heights in isogeny classes of motives\nby Ali
ce Lin (Harvard University) as part of Harvard number theory seminar\n\nLe
cture held in Science Center Room 507.\n\nAbstract\nUsing integral p-adic
Hodge theory\, Kato and Koshikawa define a generalization of the Faltings
height of an abelian variety to motives defined over a number field. Assum
ing the adelic Mumford-Tate conjecture\, we prove a finiteness property fo
r heights in the isogeny class of a motive\, where the isogenous motives a
re not required to be defined over the same number field. This expands on
a result of Kisin and Mocz for the Faltings height in isogeny classes of a
belian varieties.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi (Boston College)
DTSTART:20251008T190000Z
DTEND:20251008T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/139
DESCRIPTION:Title: A new approach to $p$-Hecke correspondences and Rapoport-Zink s
paces\nby Keerthi Madapusi (Boston College) as part of Harvard number
theory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nHe
cke operators play a fundamental role in understanding the arithmetic prop
erties of modular and automorphic forms. Since the advent of the original
Eichler-Shimura relation\, it has been clear that the mod-p behavior of He
cke correspondences is crucial for such applications. However\, one could
argue a truly robust theory of such correspondences yielding convenient ac
cess to their mod-p reductions has so far been elusive\, especially when d
ealing with higher rank groups. \n\nIn this talk\, I will present a new ap
proach to these matters\, using recent advances in p-adic geometry and p-a
dic cohomology\, building on work of Drinfeld and Bhatt-Lurie\, and combin
ing them with a tool familiar to the geometric Langlands and representatio
n theory community: the Vinberg monoid. In particular\, this approach yiel
ds direct access to geometric incarnations of the 'standard' basis element
s of the spherical Hecke algebra.\n\nFor another application\, this approa
ch also gives the first general construction of Rapoport-Zink spaces assoc
iated with exceptional groups. \n\nThis work is joint with Si Ying Lee.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brandon Alberts (Eastern Michigan University)
DTSTART:20251015T190000Z
DTEND:20251015T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/140
DESCRIPTION:Title: Recent progress towards Malle's conjecture\nby Brandon Albe
rts (Eastern Michigan University) as part of Harvard number theory seminar
\n\nLecture held in Science Center Room 507.\n\nAbstract\nMalle's conjectu
re concerns the asymptotic number of $G$-extensions with bounded discrimin
ant. We will discuss some of the more recent results in this direction\, i
ncluding inductive methods\, multivariable Dirichlet series\, and a ''twis
ted'' version of Malle's conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Sutherland (MIT)
DTSTART:20251022T190000Z
DTEND:20251022T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/141
DESCRIPTION:Title: Murmurations for elliptic curves ordered by height\nby Andr
ew Sutherland (MIT) as part of Harvard number theory seminar\n\nLecture he
ld in Science Center Room 507.\n\nAbstract\nWhile conducting machine learn
ing experiments in 2022\, He-Lee-Oliver-Pozdnyakov noticed a curious oscil
lation (murmuration) in averages of Frobenius traces of elliptic curves ov
er Q of particular ranks in prescribed conductor ranges. Similar oscillati
ons have since been observed in many other families of L-functions. For L-
functions of Hecke eigenforms with trivial character\, Zubrilina used the
Eichler-Selberg trace formula to derive a density function that completely
explains the murmuration phenomenon in this setting. Zubrilina's methods
have since been applied in other settings where a suitable trace formula i
s available\, but an explanation for the murmurations originally observed
in elliptic curves has remained\nelusive.\n\nIn this talk I will present j
oint work with Will Sawin (arXiv:2504.12295) in which we use the Voronoi s
ummation formula to analyze murmurations in the elliptic curve setting. W
e order elliptic curves by height and average against a smooth test functi
on\, which allows us to obtain an unconditional result. This leads to an
explicit murmuration density function that we conjecture applies more gene
rally and explains the original murmuration phenomenon observed by He-Lee-
Oliver-Pozdnyakov\, in which elliptic curves are ordered by\nconductor rat
her than height.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacksyn Bakeberg (Boston University)
DTSTART:20251029T190000Z
DTEND:20251029T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/142
DESCRIPTION:Title: Excursion functions on $p$-adic $\\mathrm{SL}_2$\nby Jacksy
n Bakeberg (Boston University) as part of Harvard number theory seminar\n\
nLecture held in Science Center Room 507.\n\nAbstract\nThe Bernstein cente
r of a $p$-adic group is a commutative ring of certain distributions on th
e group\, and it interacts closely with the group’s representation theor
y. Fargues and Scholze provide an abstract construction of a class of elem
ents of the Bernstein center called excursion operators\, which encode a c
andidate for the (semisimplified) local Langlands correspondence. In this
talk\, I will present an approach to understanding excursion operators con
cretely as distributions on the group\, with a special emphasis on the cas
e of $G = \\mathrm{SL}_2$ where everything can be made quite explicit. In
the pre-talk\, I will provide a gentle introduction to the Bernstein cente
r and the local Langlands correspondence for $\\mathrm{SL}_2$.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eran Assaf (MIT)
DTSTART:20251105T200000Z
DTEND:20251105T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/143
DESCRIPTION:Title: Tropicalizations of locally symmetric varieties\nby Eran As
saf (MIT) as part of Harvard number theory seminar\n\nLecture held in Scie
nce Center Room 507.\n\nAbstract\nWe relate the top-weight rational cohomo
logy of a locally symmetric variety to the cohomology of arithmetic groups
associated to its rational boundary components. This relation is given in
terms of a fundamental spectral sequence\, whose applications to the coho
mology of Siegel modular varieties and unitary modular varieties will be p
resented. By studying the combinatorics of the boundary\, we are able to e
xhibit a Hopf algebra structure\, with applications to the cohomology of a
rithmetic groups.\n\nThis is joint work with Madeline Brandt\, Juliette Br
uce\, Melody Chan and Raluca Vlad.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Santiago Arango-Piñeros (UMass Amherst)
DTSTART:20251112T200000Z
DTEND:20251112T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/144
DESCRIPTION:Title: Counting primitive integral solutions to generalized Fermat equ
ations\nby Santiago Arango-Piñeros (UMass Amherst) as part of Harvard
number theory seminar\n\nLecture held in Science Center Room 507.\n\nAbst
ract\nLet \\( F \\colon A x^a + B y^b + C z^c = 0 \\) be a generalized Fer
mat equation with\nnonzero integer coefficients. A solution \\( (x\, y\, z
) \\in \\mathbb{Z}^3\\) is called \\(\\textit{primitive}\\) if\n\\( \\gcd(
x\, y\, z) = 1 \\). We prove that when\n\\( \\chi = \\tfrac{1}{a} + \\tfra
c{1}{b} + \\tfrac{1}{c} - 1 > 0 \\)\,\nthe counting function \\( N(F\; h)
\\) of primitive integral solutions of height at most\n\\( h \\) satisfies
\n\\[\nN(F\; h) \\sim \\kappa(F) \\cdot h^{\\chi}\,\n\\]\nfor some constan
t \\( \\kappa(F) \\ge 0 \\)\, as \\( h \\to \\infty \\). This result\nrefi
nes a theorem of Beukers\, and the proof relies on the stack-theoretic\npe
rspective introduced by Poonen--Schaefer--Stoll in their study of\n\\( x^2
+ y^3 + z^7 = 0 \\).\n\nDuring the pre-talk\, I will introduce torsors an
d quotient stacks.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikayel Mkrtchyan (MIT)
DTSTART:20251119T200000Z
DTEND:20251119T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/145
DESCRIPTION:Title: Higher Siegel-Weil formula for unitary groups over function fie
lds: case of corank-1 coefficients\nby Mikayel Mkrtchyan (MIT) as part
of Harvard number theory seminar\n\nLecture held in Science Center Room 5
07.\n\nAbstract\nThe arithmetic Siegel-Weil formula relates degrees of spe
cial cycles on Shimura varieties to derivatives of certain Eisenstein seri
es. In their seminal work\, Feng-Yun-Zhang have defined analogous special
cycles on moduli spaces of shtukas over function fields\, and proved a hig
her Siegel-Weil formula relating degrees of special cycles on moduli space
s of shtukas with r legs\, to r-th derivatives of non-degenerate Fourier c
oefficients of the Eisenstein series. In this talk\, I will report on join
t work with Tony Feng and Benjamin Howard\, where we prove a higher Siegel
-Weil formula for corank-1 singular Fourier coefficients. A key feature of
the proof is an unexpected full support property of the relevant "Hitchin
" fibration.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinyu Zhou (Boston University)
DTSTART:20251203T200000Z
DTEND:20251203T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/146
DESCRIPTION:Title: Pro-étale quasicoherent cohomology of negative Banach--Colmez
spaces\nby Xinyu Zhou (Boston University) as part of Harvard number th
eory seminar\n\nLecture held in Science Center Room 507.\n\nAbstract\nNega
tive Banach--Colmez spaces are moduli spaces of extensions of vector bundl
es on the Fargues--Fontaine curve. They play important roles in the Fargue
s--Scholze program and $p$-adic local Langlands. In this talk\, I will dis
cuss how to use the newly developed 6-functor formalism of pro-étale quas
icoherent cohomology to compute the cohomology of negative Banach--Colmez
spaces. Along this way\, I will also show some tools such as Drinfeld's le
mma which are of more general interest. This is based on the joint work wi
th people from last year's AIM workshop on chromatic homotopy theory and $
p$-adic geometry. \n\nIn the pretalk\, I will give an introduction to pro-
étale cohomology and to Poincaré duality for rigid-analytic spaces.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Zhang (MIT)
DTSTART:20260204T200000Z
DTEND:20260204T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/148
DESCRIPTION:Title: A Lie-theoretic trichotomy in Diophantine geometry and arithmet
ic dynamics\nby Robin Zhang (MIT) as part of Harvard number theory sem
inar\n\nLecture held in Science Center Room 507 (Room 530 for pretalks).\n
\nAbstract\nHow can the finite/infinite dichotomy of the Killing–Cartan
classification of simple Lie groups & algebras appear in number theory? I
will explain how this Lie-theoretic dichotomy is realized in the finitenes
s or infinitude of positive integer solutions to certain Diophantine equat
ions\, and explore some of its implications for classical questions studie
d by Gauss\, Mordell\, Coxeter\, Conway\, and Schinzel in combinatorics an
d number theory. I will then switch gears to the arithmetic dynamics of cl
uster Donaldson–Thomas transformations\, which refines the Diophantine r
ealization of the finite/infinite dichotomy into a finite/affine/indefinit
e trichotomy that matches the Kac–Moody classification of infinite-dimen
sional Lie algebras.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Harvard)
DTSTART:20260211T200000Z
DTEND:20260211T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/149
DESCRIPTION:Title: Bhargava's conjecture over function fields\nby Aaron Landes
man (Harvard) as part of Harvard number theory seminar\n\nLecture held in
Science Center Room 507 (Room 232 for pretalks).\n\nAbstract\nBhargava's c
onjecture predicts the number of degree d extensions of $\\mathbb Q$. In
joint work with Ishan Levy\, we prove a version of this conjecture over $\
\mathbb F_q(t)$\, for $q$ sufficiently large relative to $d$ and prime to
$d!$. The key new input is a refined understanding of the stable homology
of Hurwitz spaces\, and more generally an understanding of the stable homo
logy of Hurwitz space modules. Time permitting\, we may also describe how
these ideas can also be used to compute the average size of Selmer groups
in quadratic twist families of elliptic curves over function fields.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zureick-Brown (Amherst College)
DTSTART:20260218T200000Z
DTEND:20260218T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/150
DESCRIPTION:Title: $\\ell$-adic Images of Galois for Elliptic Curves over $\\mathb
b Q$\nby David Zureick-Brown (Amherst College) as part of Harvard numb
er theory seminar\n\nLecture held in Science Center Room 507 (Room 232 for
pretalks).\n\nAbstract\nI will discuss recent joint work with Jeremy Rous
e and Drew Sutherland on Mazur’s “Program B” — the classification
of the possible “images of Galois” associated to an elliptic curve (eq
uivalently\, classification of all rational points on certain modular curv
es $X_H$). The main result is a provisional classification of the possible
images of l-adic Galois representations associated to elliptic curves ove
r Q and is provably complete barring the existence of unexpected rational
points on modular curves associated to the normalizers of non-split Cartan
subgroups and two additional genus 9 modular curves of level 49.\n\nI wil
l also discuss the framework and various applications (for example: a very
fast algorithm to rigorously compute the l-adic image of Galois of an ell
iptic curve over Q)\, and then highlight several new ideas from the joint
work\, including techniques for computing models of modular curves and nov
el arguments to determine their rational points\, a computational approach
that works directly with moduli and bypasses defining equations\, and (wi
th John Voight) a generalization of Kolyvagin’s theorem to the modular c
urves we study.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Swaminathan (Harvard)
DTSTART:20260225T200000Z
DTEND:20260225T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/151
DESCRIPTION:Title: Second moments for 2-Selmer structures on elliptic curves\, and
applications\nby Ashvin Swaminathan (Harvard) as part of Harvard numb
er theory seminar\n\nLecture held in Science Center Room 507 (Room 232 for
pretalks).\n\nAbstract\nA key prediction of the Poonen--Rains heuristics
is that every nonnegative integer $r$ occurs as a 2-Selmer rank for a posi
tive proportion of elliptic curves over $\\mathbb{Q}$\, but this predictio
n was not previously known for any $r$. In this talk\, we prove that a pos
itive proportion of elliptic curves over $\\mathbb{Q}$ have 2-Selmer rank
$r$\, for small values of $r$.\n\nThis talk is based on joint works with M
anjul Bhargava\, Wei Ho\, Arul Shankar\, and Ari Shnidman.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (Boston University)
DTSTART:20260304T200000Z
DTEND:20260304T210000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/152
DESCRIPTION:Title: Covers of curves\, Ceresa cycles\, and unlikely intersections\nby Padmavathi Srinivasan (Boston University) as part of Harvard number
theory seminar\n\nLecture held in Science Center Room 507 (Room 232 for p
retalks).\n\nAbstract\nThe Ceresa cycle is a canonical homologically trivi
al algebraic cycle associated to a curve in its Jacobian. In his 1983 thes
is\, Ceresa showed that this cycle is algebraically nontrivial for a very
general complex curve of genus at least 3. In the last few years\, there h
ave been many new results shedding light on the locus in the moduli space
of genus g curves where the Ceresa cycle becomes torsion. We will survey t
hese recent results and provide new examples of families of curves where o
nly finitely many members of the family have torsion Ceresa cycle. The mai
n idea is to leverage the covering map to reduce the question of torsionne
ss of the Ceresa cycle to the torsionness of a canonical point on the Jaco
bian and combine this with recent results on unlikely intersections in abe
lian varieties (the relative Manin--Mumford conjecture). This is joint wor
k with Tejasi Bhatnagar\, Sheela Devadas and Toren D'Nelly Warady.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Hamann (Harvard)
DTSTART:20260311T190000Z
DTEND:20260311T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/153
DESCRIPTION:Title: Comparison of the analytic and algebraic categorical local Lang
lands correspondences\nby Linus Hamann (Harvard) as part of Harvard nu
mber theory seminar\n\nLecture held in Science Center Room 507 (Room 232 f
or pretalks).\n\nAbstract\nWe prove a folklore conjecture identifying the
two known candidates for the automorphic side of the categorical local Lan
glands correspondence\, allowing the passage of ideas and results from one
side to the other. Precisely\, for G a connected reductive group\, we con
struct an equivalence between the derived category of etale sheaves on the
algebraic stack Isoc_{G} of G-isocrystals constructed by Zhu and the deri
ved category of étale sheaves on the analytic moduli stack of G-bundles B
un_{G} on the Fargues--Fontaine curve constructed by Fargues--Scholze. To
a (very crude) first approximation\, this is accomplished by considering a
n explicit geometric object\, denoted Bun_{G}^{mer}\, which defines a corr
espondence between the analytification of the algebraic object Isoc_{G} an
d the analytic object Bun_{G}\, and then pushing and pulling along this co
rrespondence. The resulting functor can be roughly thought of as "nearby c
ycles" between the generic and special fiber of the formal scheme (or rath
er its generalization to kimberlites in the sense of Gleason) Bun_{G}^{mer
}. In usual formal/adic geometry\, we know that such nearby cycles functor
s allow us to compare cohomology on the rigid generic fiber and special fi
ber of the formal scheme via showing that the formal scheme is henselian a
long the analytic locus coming from the rigid generic fiber. We prove our
functor is an equivalence by verifying such henselianity properties hold i
nside the space Bun_{G}^{mer}. In particular\, under our functor\, this he
nselianity property allows us to compare a natural excision filtration (or
semi-orthogonal decomposition) on the category attached to Isoc_{G} with
an "exotic" one on the category Bun_{G} coming from the existence of certa
in exceptional adjoints to the usual six operations. This reduces us to sh
owing our functor is an equivalence on the induced functor on the graded p
ieces of this filtration\, where it is easily checked to be true. This is
joint work with Ian Gleason\, Joao Lourenco\, Alexander Ivanov\, and Konra
d Zou.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (MIT)
DTSTART:20260325T190000Z
DTEND:20260325T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/154
DESCRIPTION:Title: Galois action on higher etale homotopy groups\nby Alexander
Petrov (MIT) as part of Harvard number theory seminar\n\nLecture held in
Science Center Room 507 (Room 232 for pretalks).\n\nAbstract\nTo an algebr
aic variety over a number field F one can associate its Q_p-etale cohomolo
gy groups\, equipped with an action of the absolute Galois group of F -- s
uch representations are known to enjoy several special properties that do
not hold for arbitrary representations. For example\, they are de Rham at
p and the eigenvalues of Frobenius elements at almost all places are Weil
numbers. Analogous facts hold for linear representations of the Galois gro
up that can be extracted (e.g. by considering regular functions on the pro
-algebraic completion) form the Galois action on the etale fundamental gro
up\, and one expects that all such representations arise from cohomology o
f algebraic varieties. In this talk\, I will discuss a family of examples
showing that the analogous expectation cannot hold for higher etale homoto
py groups. In particular\, one finds that (dual of) 2nd etale homotopy gro
up of the moduli space of abelian varieties of dimension g>1 contains a su
brepresentations that is not de Rham at p. This talk is based on joint wor
ks with Lue Pan and George Pappas.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juanita Duque-Rosero (Boston University)
DTSTART:20260401T190000Z
DTEND:20260401T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/155
DESCRIPTION:Title: Triangular modular curves\nby Juanita Duque-Rosero (Boston
University) as part of Harvard number theory seminar\n\nLecture held in Sc
ience Center Room 507 (Room 232 for pretalks).\n\nAbstract\nTriangular mod
ular curves are a generalization of modular curves and arise from quotient
s of the complex upper half-plane by congruence subgroups of hyperbolic tr
iangle groups. They are connected to Darmon’s program for rational point
s on generalized Fermat equations. In this talk\, we will focus on arithm
etic properties of the Borel-kind triangular modular curves and potential
applications to Darmon’s program. This is joint work with John Voight.\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART:20260408T190000Z
DTEND:20260408T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/156
DESCRIPTION:by Jennifer Balakrishnan (Boston University) as part of Harvar
d number theory seminar\n\nLecture held in Science Center Room 507 (Room 2
32 for pretalks).\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katharine Woo (Stanford University)
DTSTART:20260415T190000Z
DTEND:20260415T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/157
DESCRIPTION:by Katharine Woo (Stanford University) as part of Harvard numb
er theory seminar\n\nLecture held in Science Center Room 507 (Room 232 for
pretalks).\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (Brown University)
DTSTART:20260422T190000Z
DTEND:20260422T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/158
DESCRIPTION:by Isabel Vogt (Brown University) as part of Harvard number th
eory seminar\n\nLecture held in Science Center Room 507 (Room 232 for pret
alks).\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niven Achenjang (Harvard)
DTSTART:20260429T190000Z
DTEND:20260429T200000Z
DTSTAMP:20260404T095653Z
UID:HarvardNT/159
DESCRIPTION:by Niven Achenjang (Harvard) as part of Harvard number theory
seminar\n\nLecture held in Science Center Room 507 (Room 232 for pretalks)
.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/HarvardNT/159/
END:VEVENT
END:VCALENDAR