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BEGIN:VEVENT
SUMMARY:Ashay Burungale
DTSTART:20231017T190000Z
DTEND:20231017T200000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/3
DESCRIPTION:Title: Zeta elements for elliptic curves and some applications\nby Asha
y Burungale as part of BC-MIT number theory seminar\n\nLecture held in Mal
oney 560 at Boston College.\n\nAbstract\nThe talk plans to outline the exi
stence of two-variable zeta element over an imaginary quadratic field for
an elliptic curve defined over Q. Its arithmetic consequences include proo
f of Kobayashi's main conjecture for semistable curves and special cases o
f the Birch--Swinnerton-Dyer conjecture. (Joint with C. Skinner\, Y. Tian
and X. Wan.)\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Chan (University of Michigan)
DTSTART:20231017T203000Z
DTEND:20231017T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/4
DESCRIPTION:Title: Supercuspidal representations and very regular elements\nby Char
lotte Chan (University of Michigan) as part of BC-MIT number theory semina
r\n\nLecture held in Maloney 560 at Boston College.\n\nAbstract\nIn the 19
90s\, Henniart proved that certain supercuspidal\nrepresentations of p-adi
c GLn are characterized by their character\nvalues on very regular element
s\, a special class of regular semisimple\nelements on which character for
mulae are remarkably simple. Henniart's\nresult has seen many interesting
applications---for example\, in\ndetermining algebraic descriptions of geo
metrically arising\nrepresentations. In this talk\, we'll discuss a genera
lization of\nHenniart's theorem to general G. As a byproduct of our method
s\, we\nobtain an easy\, non-cohomological condition distinguishing unipot
ent\nsupercuspidal representations\, yielding a p-adic analogue of Lusztig
's\ncriterion for finite fields. This is joint work with M. Oi.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Peluse (University of Michigan)
DTSTART:20231114T200000Z
DTEND:20231114T210000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/5
DESCRIPTION:Title: Divisibility of character values of the symmetric group\nby Sara
h Peluse (University of Michigan) as part of BC-MIT number theory seminar\
n\nLecture held in MIT room 4-163.\n\nAbstract\nIn 2017\, Miller computed
the character tables of $S_n$ for all $n$ up to $38$ and looked at various
statistical properties of the entries. Characters of symmetric groups tak
e only integer values\, and\, based on his computations\, Miller conjectur
ed that almost all entries of the character table of $S_n$ are divisible b
y any fixed prime power as $n$ tends to infinity. In this talk\, I will di
scuss joint work with K. Soundararajan that resolves this conjecture\, and
mention some related open problems.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samit Dasgupta (Duke University)
DTSTART:20231114T213000Z
DTEND:20231114T223000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/6
DESCRIPTION:Title: On the Brumer-Stark conjecture and refinements\nby Samit Dasgupt
a (Duke University) as part of BC-MIT number theory seminar\n\nLecture hel
d in MIT room 2-449.\n\nAbstract\nIn this talk I will describe my recent w
ork with Mahesh Kakde on the Brumer-Stark Conjecture and certain refinemen
ts. I will give a broad overview that motivates the conjecture and gives c
onnections to explicit class field theory. I will conclude with a descript
ion of recent work (joint w/ Kakde\, Jesse Silliman\, and Jiuya Wang) in w
hich we complete the proof of the conjecture. Moreover\, we deduce a certa
in special case of the Equivariant Tamagawa Number Conjecture\, which has
important corollaries. The key aspect of the most recent results\, which a
llows us to handle the prime $p=2$\, is the proof of a version of Ribet's
Lemma in the case of characters that are congruent modulo $p$.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander D. Smith (UCLA)
DTSTART:20240409T190000Z
DTEND:20240409T200000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/7
DESCRIPTION:Title: Simple abelian varieties over finite fields with extreme point count
s\nby Alexander D. Smith (UCLA) as part of BC-MIT number theory semina
r\n\nLecture held in Maloney 560 at Boston College.\n\nAbstract\nGiven a c
ompactly supported probability measure on the reals\, we will give a neces
sary and sufficient condition for there to be a sequence of totally real a
lgebraic integers whose distribution of conjugates approaches the measure.
We use this result to prove that there are infinitely many totally positi
ve algebraic integers X satisfying tr(X)/deg(X) < 1.899\; previously\, the
re were only known to be infinitely many such integers satisfying tr(X)/de
g(X) < 2. We also will explain how our method can be used in the search fo
r simple abelian varieties with extreme point counts.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bennett (University of British Columbia)
DTSTART:20240409T203000Z
DTEND:20240409T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/8
DESCRIPTION:Title: Arithmetic progressions in sumsets of geometric progressions\nby
Michael Bennett (University of British Columbia) as part of BC-MIT number
theory seminar\n\nLecture held in Maloney 560 at Boston College.\n\nAbstr
act\nIf A and B are two geometric progressions\, we characterize all 3-ter
m arithmetic progressions in the sumset A+B. Somewhat surprisingly\, while
mostly elementary\, this appears to require quite deep machinery from Dio
phantine Approximation.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yiannis Sakellaridis (Johns Hopkins University)
DTSTART:20240319T190000Z
DTEND:20240319T200000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/9
DESCRIPTION:Title: Relative Langlands duality\, past and future\nby Yiannis Sakella
ridis (Johns Hopkins University) as part of BC-MIT number theory seminar\n
\nLecture held in MIT room 2-449.\n\nAbstract\nSince Riemann's 1859 report
on the zeta function\, it is known that certain automorphic $L$-functions
can be represented as ("period") integrals\, which often proves analytic
properties such as the functional equation. The method was advanced by Jac
quet\, Piatetski-Shapiro\, Rallis\, and many others since the 1970s\, givi
ng rise to the "relative" Langlands program. It turns out that the relatio
nship between periods and $L$-functions reflects a duality between certain
Hamiltonian varieties for a reductive group and its Langlands dual group.
I will set up this duality in a limited setting (joint work with David Be
n-Zvi and Akshay Venkatesh)\, and speculate on how it might be expanded in
the future.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Ben-Zvi (University of Texas at Austin)
DTSTART:20240319T203000Z
DTEND:20240319T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/10
DESCRIPTION:Title: Geometric Arthur parameters\nby David Ben-Zvi (University of Te
xas at Austin) as part of BC-MIT number theory seminar\n\nLecture held in
MIT room 2-449.\n\nAbstract\nArthur proposed a description of automorphic
forms in terms of tempered automorphic forms for centralizers of SL2 homom
orphisms. I will explain a point of view on the Arthur parameterization in
the setting of function fields coming from relative Langlands duality\, e
mphasizing the role of $shearing$ (the symmetry of the derived category of
graded vector spaces which simultaneously shifts weights and cohomologica
l degrees). Shearing helps account for a deficit of Tannakian formalism in
the derived context - there are more eigenvalues for tensor actions than
one might expect. The talk reports on joint work with Yiannis Sakellaridis
and Akshay Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (Berkeley)
DTSTART:20240514T190000Z
DTEND:20240514T200000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/11
DESCRIPTION:Title: The arithmetic of power series and applications to irrationality\nby Yunqing Tang (Berkeley) as part of BC-MIT number theory seminar\n\nL
ecture held in Maloney 560 at Boston College.\n\nAbstract\nIn this talk\,
we will discuss various irrationality and linear independence problems inc
luding certain products of two (classical or p-adic) log values. The proof
s use an arithmetic holonomicity theorem\, the special case of which was u
sed in the proof of the unbounded denominators conjecture\; our arithmetic
holonomicity theorem is inspired from Andre’s work on Grothendieck-Katz
p-curvature conjecture on arithmetic differential equations. A geometric
version of our arithmetic holonomicity theorems have also been studied in
recent work of Bost and Charles.\n\nThis is joint work in progress with Fr
ank Calegari and Vesselin Dimitrov.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Shnidman (Hebrew University of Jerusalem)
DTSTART:20240514T203000Z
DTEND:20240514T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/12
DESCRIPTION:Title: Vanishing criteria for Ceresa cycles\nby Ari Shnidman (Hebrew U
niversity of Jerusalem) as part of BC-MIT number theory seminar\n\nLecture
held in Maloney 560 at Boston College.\n\nAbstract\nThe Ceresa cycle of a
curve is perhaps the simplest example of a\nhomologically trivial algebra
ic cycle which need not be algebraically\ntrivial. Its vanishing in the Ch
ow (resp. Griffiths) group has various\nimplications\, but the locus of va
nishing Ceresa cycles in $M_g$ is quite\nmysterious\, beyond the fact that
it contains the hyperelliptic locus. I'll\npresent new vanishing criteria
for the Ceresa cycle of curves with\nautomorphisms\, one of them conditio
nal on the Hodge conjecture. In certain\nlow genus cases the relevant Hodg
e conjecture is known\, and using this we\ndescribe the locus of Picard cu
rves with vanishing Ceresa cycle. This is\njoint work with Jef Laga.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Lemke Oliver (Tufts University)
DTSTART:20241029T190000Z
DTEND:20241029T200000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/13
DESCRIPTION:Title: Enumerating Galois extensions of number fields\nby Robert Lemke
Oliver (Tufts University) as part of BC-MIT number theory seminar\n\nLect
ure held in Room 2-449 at MIT.\n\nAbstract\nLet $k$ be a number field. We
provide an asymptotic formula for the number of Galois extensions of $k$ w
ith absolute discriminant bounded by some $X \\geq 1$ as $X \\to \\infty$.
We also provide an asymptotic formula for the closely related count of ex
tensions $K/k$ whose normal closure has discriminant bounded by $X$. The k
ey behind these results is a new upper bound on the number of Galois exten
sions of $k$ with a given Galois group $G$ and discriminant 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 particular\, ours is the
first bound for general $G$ with an exponent that decays as $|G| \\to \\i
nfty$.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hang Xue (University of Arizona)
DTSTART:20241029T203000Z
DTEND:20241029T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/14
DESCRIPTION:Title: Fourier--Jacobi periods on unitary groups\nby Hang Xue (Univers
ity of Arizona) as part of BC-MIT number theory seminar\n\nLecture held in
Room 2-449 at MIT.\n\nAbstract\nWe explain a proof of the Gan--Gross--Pra
sad conjecture for Fourier--Jacobi periods on unitary groups via relative
trace formulae. This is joint work with Paul Boisseau and Weixiao Lu.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Patrikis (Ohio State University)
DTSTART:20241203T200000Z
DTEND:20241203T210000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/15
DESCRIPTION:Title: Compatibility of canonical l-adic local systems on Shimura varietie
s of non-abelian type\nby Stefan Patrikis (Ohio State University) as p
art of BC-MIT number theory seminar\n\nLecture held in Maloney 560 at Bost
on College.\n\nAbstract\nLet $(G\, X)$ be a Shimura datum\, and let $K$ be
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\; unlike in abelian type (moduli of abe
lian varieties\, etc.)\, in non-abelian type this problem remains complete
ly mysterious. I will discuss joint work with Christian Klevdal showing th
at for "superrigid\," including all non-abelian type\, Shimura varieties t
he points (over number fields\, say) at least yield compatible systems of
l-adic representations\, which should be the l-adic realizations of the co
njectural motives. Time permitting\, I will discuss some work in progress
(with Jake Huryn\, Kiran Kedlaya\, and Klevdal) on a crystalline analogue.
\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Myrto Mavraki (University of Toronto)
DTSTART:20241203T213000Z
DTEND:20241203T223000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/16
DESCRIPTION:Title: Uniformity in unlikely intersections and the dynamical André--Oort
conjecture\nby Myrto Mavraki (University of Toronto) as part of BC-MI
T number theory seminar\n\nLecture held in Maloney 560 at Boston College.\
n\nAbstract\nA rational map is postcritically finite (PCF) if its critical
orbits are finite. Postcritically finite maps play an important role in d
ynamics. Further\, it was suggested by Silverman that they play a role ana
logous to CM elliptic elliptic curves. Inspired in part by the Pink-Zilber
conjectures in unlikely intersections\, Baker and DeMarco formulated a co
njecture aiming to describe the subvarieties of $M_d$ that contain a Zaris
ki dense set of PCF points. Their conjecture\, now known as dynamical Andr
é--Oort conjecture (or DAO)\, was recently resolved in the case of curves
by Ji--Xie\, but remains open in higher dimensions. In this talk we will
describe recent work with DeMarco and Ye\, providing uniform bounds on the
configurations of PCF points in families of subvarieties in $M_d$. We als
o provide a gap principle in the spirit of Dimitrov--Gao--Habegger's\, Kü
hne's\, and Gao--Ge--Kühne's work on the uniform Mordell--Lang conjecture
.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chen Wan (Rutgers University)
DTSTART:20250225T200000Z
DTEND:20250225T210000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/17
DESCRIPTION:Title: Some spherical varieties of Whittaker type\nby Chen Wan (Rutger
s University) as part of BC-MIT number theory seminar\n\nLecture held in R
oom 2-449 at MIT.\n\nAbstract\nIn this talk I will discuss a special categ
ory of spherical varieties whose L^2 space is exactly supported on all the
tempered generic representations. Then I will give two families of exampl
es\, one from the theory of double flag varieties\, and the other one from
the theory of Rankin-Selberg integrals. I will also explain how to use th
is theory to prove the local funcational equations for some Rankin-Selberg
integrals. This is an ongoing joint work with Lei Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tonghai Yang (University of Wisconsin\, Madison)
DTSTART:20250225T213000Z
DTEND:20250225T223000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/18
DESCRIPTION:Title: (A)FL at infinity and arithmetic generating series of CM cycles
\nby Tonghai Yang (University of Wisconsin\, Madison) as part of BC-MIT nu
mber theory seminar\n\nLecture held in Room 2-449 at MIT.\n\nAbstract\nIn
this talk\, we propose a FL and AFL at the real place with a proof of FL (
if time permits). We also define a generating series of arithmetic CM cycl
es indexed by integer and conjecture it to be modular. Finally\, we explai
n the connection between the two. This is a preliminary report of my joint
work with Andreas Mihatsch and Siddarth Sankaran.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor Wooley (Purdue University)
DTSTART:20250401T190000Z
DTEND:20250401T200000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/19
DESCRIPTION:Title: Optimal mean value estimates and function field arithmetic\nby
Trevor Wooley (Purdue University) as part of BC-MIT number theory seminar\
n\nLecture held in Maloney 560 at Boston College.\n\nAbstract\nEssentially
optimal estimates have been obtained for mean values of Vinogradov’s ex
ponential sum as a consequence of the decoupling method (by Bourgain\, Dem
eter and Guth)\, and the efficient congruencing method (by the speaker). S
uch work makes essential use of the fact that the system of Diophantine eq
uations associated with these mean values is translation-dilation invarian
t. We report on progress for systems which are not translation-dilation in
variant obtained by exploiting the arithmetic of function fields over the
field of p-adic numbers.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Tsimerman (University of Toronto)
DTSTART:20250401T203000Z
DTEND:20250401T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/20
DESCRIPTION:Title: Good reductions of CM points for Exceptional Shimura Varieties\
nby Jacob Tsimerman (University of Toronto) as part of BC-MIT number theor
y seminar\n\nLecture held in Maloney 560 at Boston College.\n\nAbstract\nG
iven an Elliptic curve E with complex multiplication\, it is known that E
has (potentially) good reduction everywhere. Concretely\, this means that
the j-invariant of E is an algebraic integer. The generalization of this r
esult to Abelian-Varieties follows from the Neron-Ogg-Shafarevich criterio
n for good reduction.\n\nWe generalize this result to Exceptional Shimura
varieties S. Concretely\, we show that there exists some integral model S_
0 of S such that all special points of S extend to integral points of S_0.
To prove this we establish a Neron-Ogg-Shafarevich criterion in this sett
ing. Our methods are general and apply\, in particular\, to arbitrary vari
ations of hodge structures with an immersive Kodaira-Spencer map.\n\nWe wi
ll explain the proof (which is largely in the realm of birational p-adic g
eometry) and the open questions that remain. This is joint work with Ben B
akker.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Dunn (Georgia Institute of Technology)
DTSTART:20250506T193000Z
DTEND:20250506T203000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/21
DESCRIPTION:Title: Recent progress on Gauss sums and primes\nby Alexander Dunn (Ge
orgia Institute of Technology) as part of BC-MIT number theory seminar\n\n
Lecture held in Room 2-449 at MIT.\n\nAbstract\nLarge sieve inequalities a
re a fundamental tool used to investigate prime numbers and exponential su
ms. In this lecture I will explain my work that resolves a 1978 conjecture
of S. Patterson (conditional on the Generalized Riemann hypothesis) conce
rning the bias of cubic Gauss sums over the prime numbers. This explains a
well-known numerical bias first observed by Kummer in 1846. This bias was
later the subject of testing on some of the first super computers in the
20th century. Time permitting\, results on higher order Gauss sums will be
discussed. This is joint work with Maksym Radziwill.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lue Pan (Princeton University)
DTSTART:20250506T180000Z
DTEND:20250506T190000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/22
DESCRIPTION:Title: Plectic Lie algebra action on the cohomology of Hilbert modular var
ieties\nby Lue Pan (Princeton University) as part of BC-MIT number the
ory seminar\n\nLecture held in Room 2-449 at MIT.\n\nAbstract\nA result of
Jan Nekovář says that the Galois action on p-adic intersection cohomolo
gy of Hilbert modular varieties with coefficients in automorphic local sys
tems is semisimple. We will explain a new proof of this result for the non
-CM part of the cohomology and extend it to the locally analytic completed
cohomology. Interestingly\, Nekovář’s approach is based on the constr
uction of partial Frobenii at places away from p\, while our method uses p
artial Sen operators at p to construct a plectic Lie algebra action (whose
meaning will be explained in the talk). This is joint work in progress wi
th Yuanyang Jiang.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katherine Stange (University of Colorado Boulder)
DTSTART:20251021T190000Z
DTEND:20251021T200000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/23
DESCRIPTION:Title: The arithmetic of thin orbits\nby Katherine Stange (University
of Colorado Boulder) as part of BC-MIT number theory seminar\n\nLecture he
ld in Maloney 560 at Boston College.\n\nAbstract\nWe consider the local-to
-global question for orbits of thin groups/semigroups. We will discuss Ap
ollonian circle packings\, continued fractions\, and some related problems
. In the Apollonian case\, we ask about the integers which occur as curva
tures in a packing. We observe that they satisfy certain congruence restr
ictions\, and ask whether all sufficiently large integers otherwise occur.
In the case of continued fractions\, we consider variants of Zaremba's c
onjecture on the rationals with bounded continued fractions. Joint work i
ncludes work with Haag\, Kertzer\, and Rickards.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniil Rudenko
DTSTART:20251021T203000Z
DTEND:20251021T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/24
DESCRIPTION:Title: Multiple polylogarithms and homology of general linear groups\n
by Daniil Rudenko as part of BC-MIT number theory seminar\n\nLecture held
in Maloney 560 at Boston College.\n\nAbstract\nGoncharov’s program is a
chain of constructions and conjectures connecting algebraic $K$-theory\, m
ultiple polylogarithms\, and mixed Tate motives. I will describe a connect
ion between multiple polylogarithms and the homology of general linear gro
ups\, and discuss its consequences for Goncharov’s program. The talk is
based on joint work with Alexander Kupers and Ismael Sierra.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ananth Shankar
DTSTART:20251118T203000Z
DTEND:20251118T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/25
DESCRIPTION:Title: $p$-adic hyperbolicity for Shimura varieties and period images\
nby Ananth Shankar as part of BC-MIT number theory seminar\n\nLecture held
in 2-449 at MIT.\n\nAbstract\nBorel proved that every holomorphic map fro
m a product of punctured unit discs to a complex Shimura variety extends t
o a map from a product of discs to its Bailey-Borel compactification. In j
oint work with Oswal\, Zhu\, and Patel\, we proved a p-adic version of thi
s theorem over discretely valued fields for Shimura varieties of abelian t
ype. I will speak about work with Bakker\, Oswal\, and Yao\, where we prov
e the analogous $p$-adic extension theorem for compact non-abelian Shimura
varieties and geometric period images for large primes $p$.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (UC Berkeley)
DTSTART:20251118T220000Z
DTEND:20251118T230000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/26
DESCRIPTION:Title: Mirror symmetry and the Breuil—Mezard Conjecture: an update\n
by Tony Feng (UC Berkeley) as part of BC-MIT number theory seminar\n\nLect
ure held in 2-449 at MIT.\n\nAbstract\nThe Breuil—Mezard Conjecture pred
icts a precise indexing of cycles in moduli spaces of local Galois represe
ntations by modular representations of finite groups of Lie type. A couple
years ago\, Bao Le Hung and I introduced a new approach to the Breuil—M
ezard Conjecture based on a connection to an instance of mirror symmetry\,
which in that instance predicts a precise indexing of Lagrangians in a sy
mplectic variety by representations of a quantum group. Recently\, we used
this to prove the Breuil—Mezard Conjecture in the generic range for arb
itrary unramified groups\, including exceptional groups. My intent is to r
eview this and also work-in-progress with Le Hung and Zhongyipan Lin\, whi
ch aims to extend the result to ramified groups. The key new aspect of the
ramified case is a nascent theory of "Spectral Langlands functoriality"\,
an analogue of Langlands functoriality for the spectral (i.e.\, "Galois")
side of the Langlands correspondence.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Pila
DTSTART:20251202T213000Z
DTEND:20251202T223000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/27
DESCRIPTION:Title: Unlikely intersections with Bessel functions and Laguerre polynomia
ls II\nby Jonathan Pila as part of BC-MIT number theory seminar\n\nLec
ture held in Maloney 560 at Boston College.\n\nAbstract\nSeveral problems
in arithmetic geometry identify a class of "special points" and seek to de
scribe the set of special points satisfying a given system of algebraic eq
uations. Two classical examples are the Manin-Mumford conjecture concernin
g torsion points in abelian varieties\, and the Andre-Oort conjecture conc
erning CM points in Shimura varieties. We propose a new variation on this
theme\, where the role of special points is played by the zeros of special
functions such as the Bessel function\, or of classical orthogonal polyno
mials. The Bessel function problem is related to a conjecture of Fuglede i
n harmonic analysis from 1974\, and the orthogonal polynomial problem to a
conjecture of Stieltjes from 1890.\n\nIn the first talk we will recall th
e Manin-Mumford conjecture and introduce the Bessel function and orthogona
l polynomial analogs. We will sketch the proof of classical Manin-Mumford
using o-minimality and how it can be adapted to obtain various results in
this new context. This naturally leads to the more "exotic" o-minimal stru
cture of multisummable functions\, as opposed to the structure R_{an\,exp}
used in classical applications.\nIn the second talk we will discuss new f
unctional transcendence results that are needed to complete the argument:
a variant of the Ax-Schanuel theorem for the special functions appearing i
n this context. This naturally leads to the study of differential Galois g
roups for irregular-singular systems\, as opposed to the regular-singular
systems used in classical applications.\n\nAll the results are based on jo
int work with Avner Kiro and some also with Gady Kozma.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gal Binyamini
DTSTART:20251202T200000Z
DTEND:20251202T210000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/28
DESCRIPTION:Title: Unlikely intersections with Bessel functions and Laguerre polynomia
ls I\nby Gal Binyamini as part of BC-MIT number theory seminar\n\nLect
ure held in Maloney 560 at Boston College.\n\nAbstract\nSeveral problems i
n arithmetic geometry identify a class of "special points" and seek to des
cribe the set of special points satisfying a given system of algebraic equ
ations. Two classical examples are the Manin-Mumford conjecture concerning
torsion points in abelian varieties\, and the Andre-Oort conjecture conce
rning CM points in Shimura varieties. We propose a new variation on this t
heme\, where the role of special points is played by the zeros of special
functions such as the Bessel function\, or of classical orthogonal polynom
ials. The Bessel function problem is related to a conjecture of Fuglede in
harmonic analysis from 1974\, and the orthogonal polynomial problem to a
conjecture of Stieltjes from 1890.\n\nIn the first talk we will recall the
Manin-Mumford conjecture and introduce the Bessel function and orthogonal
polynomial analogs. We will sketch the proof of classical Manin-Mumford u
sing o-minimality and how it can be adapted to obtain various results in t
his new context. This naturally leads to the more "exotic" o-minimal struc
ture of multisummable functions\, as opposed to the structure R_{an\,exp}
used in classical applications.\nIn the second talk we will discuss new fu
nctional transcendence results that are needed to complete the argument: a
variant of the Ax-Schanuel theorem for the special functions appearing in
this context. This naturally leads to the study of differential Galois gr
oups for irregular-singular systems\, as opposed to the regular-singular s
ystems used in classical applications.\n\nAll the results are based on joi
nt work with Avner Kiro and some also with Gady Kozma.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Skinner (Princeton University)
DTSTART:20260210T203000Z
DTEND:20260210T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/29
DESCRIPTION:Title: Euler systems and relative cohomology\nby Christopher Skinner (
Princeton University) as part of BC-MIT number theory seminar\n\nLecture h
eld in 2-449 at MIT.\n\nAbstract\nEuler systems -- organized collections o
f Galois cohomology classes for arithmetically interesting p-adic Galois r
epresentations -- have been a useful tool for establishing the conjectured
relation between special values of L-functions and the ranks and orders o
f Selmer groups when they exist. In this talk I will describe recent work
providing new examples of Euler systems with cyclotomic variation\, inclu
ding an Euler system for the symmetric square of a modular form. As a rep
lacement for the motivic origin of prior examples\, we find the Galois ext
ensions in the relative cohomology of Shimura varieties. The control neede
d to establish the norm relations and make connections with L-values is pr
ovided by recent results in integral p-adic Hodge theory\, allowing explic
it connection with holomorphic automorphic forms.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Romyar Sharifi (University of California\, Los Angeles)
DTSTART:20260210T220000Z
DTEND:20260210T230000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/30
DESCRIPTION:Title: Eisenstein cocycles for imaginary quadratic fields\nby Romyar S
harifi (University of California\, Los Angeles) as part of BC-MIT number t
heory seminar\n\nLecture held in 2-449 at MIT.\n\nAbstract\nI will discuss
the construction of maps from the homology of Bianchi spaces for an imagi
nary quadratic field F to second K-groups of ray class fields of F. These
maps are “Eisenstein” in the sense that they factor through the quoti
ent by the action of an Eisenstein ideal way from the level. They are dire
ct analogues of known explicit maps in the setting of modular curves and c
yclotomic fields. This is largely joint work with E. Lecouturier\, S. Shih
\, and J. Wang\, though I intend to motivate this through the lens of work
-in-progress on "artificial complexes" that aims to provide explicit formu
las in terms of Steinberg symbols of elliptic units\, as in the cyclotomic
setting.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adrian Diaconu
DTSTART:20260317T190000Z
DTEND:20260317T200000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/31
DESCRIPTION:Title: Moments of quadratic L-functions over function fields\nby Adria
n Diaconu as part of BC-MIT number theory seminar\n\nLecture held in Malon
ey 560 at Boston College.\n\nAbstract\nIn 2001\, Conrey\, Farmer\, Keating
\, Rubinstein\, and Snaith developed a "recipe" utilizing heuristic argume
nts to predict the asymptotics of moments of various families of $L$-funct
ions. This heuristic was later extended by Andrade and Keating to include
moments and ratios of the family of $L$-functions associated to hyperellip
tic curves of genus $g$ over a fixed finite field. In joint work with Berg
ström\, Petersen\, and Westerland\, we related the moment conjecture of A
ndrade and Keating to the problem of understanding the homology of the bra
id group with symplectic coefficients. We computed the stable homology gro
ups of the braid groups with these coefficients\, together with their stru
cture as Galois representations\, and showed that the answer matches the n
umber-theoretic predictions. Our results\, combined with a recent homologi
cal stability theorem of Miller\, Patzt\, Petersen\, and Randal-Williams\,
imply the conjectured asymptotics for all moments in the function field c
ase\, for all large enough odd prime powers $q$.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zywina
DTSTART:20260317T203000Z
DTEND:20260317T213000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/32
DESCRIPTION:Title: Elliptic curves of low rank\nby David Zywina as part of BC-MIT
number theory seminar\n\nLecture held in Maloney 560 at Boston College.\n\
nAbstract\nFor an elliptic curve $E$ over a number field $K$\, the set $E(
K)$ of $K$-points is a finitely generated abelian group whose rank is an i
mportant/mysterious invariant. It is an open and difficult problem to det
ermine which ranks occur for elliptic curves over a fixed number field $K$
. We will discuss recent work which shows that there are infinitely many e
lliptic curves over $K$ of rank $r$ for each integer $0 \\leq r \\leq 4$.
We will construct our curves by specializing well-chosen nonisotrivial f
amilies. We will use a result of Kai\, which generalizes work of Green\,
Tao and Ziegler to number fields\, to carefully choose our curves in the f
amilies.\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giada Grossi
DTSTART:20260414T193000Z
DTEND:20260414T203000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/33
DESCRIPTION:by Giada Grossi as part of BC-MIT number theory seminar\n\nLec
ture held in 2-449 at MIT.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/BC-MIT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gregorio Baldi (IMJ and IAS)
DTSTART:20260414T210000Z
DTEND:20260414T220000Z
DTSTAMP:20260404T092654Z
UID:BC-MIT/34
DESCRIPTION:Title: Intersections and the Bézout Range for Abelian Varieties\nby G
regorio Baldi (IMJ and IAS) as part of BC-MIT number theory seminar\n\nLec
ture held in 2-449 at MIT.\n\nAbstract\nGiven subvarieties X\,Y of a compl
ex algebraic variety S of complementary dimension\, must they intersect? W
hen S is projective space\, this is a consequence of the classical Bézout
theorem\, and an analogue for simple abelian varieties was established by
Barth in 1968. Moreover\, the moving lemma suggests that\, after suitable
translations\, one may arrange for intersections of the expected dimensio
n.\nIn this talk\, we describe variants for simple abelian varieties in th
e spirit of the completed Zilber--Pink philosophy. When X and Y have compl
ementary dimension\, we show that the intersections X∩[n]Y are zero-dime
nsional for all but finitely many integers n\, and that these intersection
s collectively give rise to an analytically dense subset of X as n varies.
We moreover control those n for which X∩[n]Y has a positive dimensional
component uniformly in X\,Y and A. When dimX+dimY