BEGIN:VCALENDAR
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BEGIN:VEVENT
SUMMARY:Michael Harris (Columbia University)
DTSTART:20200513T170000Z
DTEND:20200513T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/1
DESCRIPTION:Title: Local Langlands parametrization for G2\nby
Michael Harris (Columbia University) as part of Fields Number Theory Semi
nar\n\n\nAbstract\nPlease register for this talk here: https://zoom.us/mee
ting/register/tJwlduurrzMqE9PidEG3TfLJ2qHG9dEp5cua . \n\nThis is a report
on joint work with C. Khare and J. Thorne. We construct a Langlands parame
trization of supercuspidal representations of $G_2$ over a $p$-adic fields
. More precisely\, for any finite extension $K/\\mathbb{Q}_p$ we will cons
truct a bijection \n\n$\\mathcal{L}_g : \\mathcal{A}_g^0 (G_2\, K)\\righta
rrow \\mathcal{G}^0(G_2\, K)$\n\nfrom the set of generic supercuspidal rep
resentations of $G_2(K)$ to the set of irreducible continuous homomorphism
s $\\rho: W_K \\rightarrow G_2(\\mathbb{C})$ with $W_K$ the Weil group of
$K$. The construction of the map is simply a matter of assembling argument
s that are already in literature\, plus an unpublished result of Savin (in
cluded as an appendix in our article) on the global genericity of an excep
tional theta lift. The proof of surjectivity is an application of a recent
result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$. T
his allows us to carry out a strategy\, based on automorphy lifting theore
ms\, that was initially developed in our joint work with G. Böckle on pot
ential automorphy over function fields. The proof of injectivity also uses
global arithmetic methods.\n\nFor an introductory lecture on this topic\,
please see the video below: https://youtu.be/syU4h0ELK-I .\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritabrata Munshi (Tata Institute of Fundamental Research and India
n Statistical Institute)
DTSTART:20200520T170000Z
DTEND:20200520T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/2
DESCRIPTION:Title: Circle method and subconvexity\nby Ritabra
ta Munshi (Tata Institute of Fundamental Research and Indian Statistical I
nstitute) as part of Fields Number Theory Seminar\n\n\nAbstract\nPlease re
gister for this talk here: https://zoom.us/meeting/register/tJwlduurrzMqE9
PidEG3TfLJ2qHG9dEp5cua .\n\nIn this talk I will discuss the subconvexity p
roblem for GL(3)xGL(2) Rankin-Selberg L-functions. I will show how the del
ta method can be used to prove various subconvexity bounds for such L-func
tions. The talk should be accessible to graduate students.\n\nFor the intr
oductory lecture for this talk\, please see: https://youtu.be/LJeyes0BpOk
.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sujatha Ramdorai (University of British Columbia)
DTSTART:20200527T170000Z
DTEND:20200527T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/3
DESCRIPTION:Title: Euler characteristics and Arithmetic\nby S
ujatha Ramdorai (University of British Columbia) as part of Fields Number
Theory Seminar\n\n\nAbstract\nPlease register for this talk here: https://
zoom.us/meeting/register/tJwlduurrzMqE9PidEG3TfLJ2qHG9dEp5cua .\n\nThe Eul
er characteristic of the Selmer groups of elliptic curves encodes informat
ion on the arithmetic of the elliptic curve. We discuss the connection to
the Birch and Swinnerton Dyer conjecture and study the Euler characteristi
cs of residually isomorphic Galois representations.\n\nFor the introductor
y lecture to this talk\, please see: https://youtu.be/YYh9TodNPdQ .\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chandrashekhar Khare (UCLA)
DTSTART:20200603T170000Z
DTEND:20200603T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/4
DESCRIPTION:Title: Wiles defect for Hecke algebras that are not c
omplete intersections\nby Chandrashekhar Khare (UCLA) as part of Field
s Number Theory Seminar\n\n\nAbstract\nPlease register for this talk here:
https://zoom.us/meeting/register/tJwlduurrzMqE9PidEG3TfLJ2qHG9dEp5cua .\n
\nIn his work on modularity theorems\, Wiles proved a numerical criterion
for a map of rings R->T to be an isomorphism of complete intersections. In
addition to proving modularity theorems\, this numerical criterion also i
mplies a connection between the order of a certain Selmer group and a spec
ial value of an L-function.\n\nIn this talk I will consider the case of a
Hecke algebra acting on the cohomology a Shimura curve associated to a qua
ternion algebra. In this case\, one has an analogous map of rings R->T whi
ch is known to be an isomorphism\, but in many cases the rings R and T fai
l to be complete intersections. This means that Wiles's numerical criterio
n will fail to hold.\n\nI will describe a method for precisely computing t
he extent to which the numerical criterion fails (i.e. the 'Wiles defect")
at a newform f which gives rise to an augmentation T -> Z_p. The defect t
urns out to be determined entirely by local information at the primes q di
viding the discriminant of the quaternion algebra at which the mod p repre
sentation arising from f is ``trivial''. (For instance if f corresponds to
a semistable elliptic curve\, then the local defect at q is related to th
e ``tame regulator'' of the Tate period of the elliptic curve at q.)\n\nFu
rther reading: \n\nMy talk will be based on:\n\nWiles defect for Hecke alg
ebras that are not complete intersection\njoint with G. Böckle and Jeff M
anning\navailable at: https://arxiv.org/abs/1910.08507\n\nAnother referenc
e to give some context:\n\nAUTHOR = Ribet\, Kenneth A\nTITLE = Multiplicit
ies of Galois representations in Jacobians of Shimura curves\nBOOKTITLE =
Festschrift in honor of I.I. Piatetski-Shapiro on the occasion of his sixt
ieth birthday\, Part II Ramat Aviv\, 1989\nSERIES = Israel Math. Conf. Pro
c.\nVOLUME = 3\nPAGES = 221--236\nPUBLISHER = Weizmann\, Jerusalem\nYEAR =
1990\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mahesh Kakde (Indian Institute of Science)
DTSTART:20200610T170000Z
DTEND:20200610T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/5
DESCRIPTION:Title: Integral Gross-Stark conjecture and explicit f
ormulae for Brumer-Stark units\nby Mahesh Kakde (Indian Institute of S
cience) as part of Fields Number Theory Seminar\n\n\nAbstract\nPlease regi
ster for this talk here: https://zoom.us/meeting/register/tJwlduurrzMqE9Pi
dEG3TfLJ2qHG9dEp5cua .\n\nIn this talk I will present integral refinements
of the Gross-Stark conjecture\, due to Gross\, known as the tower of fiel
ds conjecture. I will then describe my on-going work proving this conjectu
re. The tower of fields conjecture implies a conjecture of Dasgupta that g
ives an explicit p-adic analytic formula for the Brumer-Stark units. I wil
l sketch this along with an application to Hilbert’s 12th problem. This
is all a joint work with Samit Dasgupta.\n\nFor the introductory lecture t
o this talk\, please see: https://youtu.be/oJIfdh1ibH4\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen Kudla (University of Toronto)
DTSTART:20200617T170000Z
DTEND:20200617T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/6
DESCRIPTION:Title: On the subring of special cycles on orthogonal
Shimura varieties\nby Stephen Kudla (University of Toronto) as part o
f Fields Number Theory Seminar\n\n\nAbstract\nPlease register for this tal
k here: https://zoom.us/meeting/register/tJwlduurrzMqE9PidEG3TfLJ2qHG9dEp5
cua .\n\nPlease refer to the introductory slides for this talk here: http:
//www.fields.utoronto.ca/sites/default/files/uploads/Fields.prep.talk.2020
.pdf . \n\nBy old results with Millson\, the generating series for the coh
omology classes of special cycles on orthogonal Shimura varieties over a t
otally real field are Hilbert-Siegel modular forms. These forms arise via
theta series. Using this result and the Siegel-Weil formula\, we show that
the products in the subring of cohomology generated by the special cycles
are controlled by the Fourier coefficients of triple pullbacks of certain
Siegel-Eisenstein series. As a consequence\, there are comparison isomorp
hisms between special subrings for different Shimura varieties that may be
of motivic origin.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henri Darmon (McGill University)
DTSTART:20200708T170000Z
DTEND:20200708T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/7
DESCRIPTION:Title: Generating series for RM values of rigid merom
orphic cocycles\nby Henri Darmon (McGill University) as part of Fields
Number Theory Seminar\n\n\nAbstract\nPlease register here: https://zoom.u
s/meeting/register/tJEtcuuvqTksHdfsLV9ZiYeWaHDdqnotA5-5 .\n\nI will descri
be two ongoing works whose unifying theme is to establish the algebraicity
of the RM values of rigid meromorphic cocycles\, by realizing these invar
iants as the fourier coefficients of certain p-adic modular generating ser
ies\, thereby obtaining the desired properties from the theory of deformat
ions of Galois representations and from global class field theory. The fir
st project\, in collaboration with Alice Pozzi and Jan Vonk\, considers th
e RM values of the Dedekind-Rademacher cocycle and its relation to the dia
gonal restrictions of certain first order deformations of Hilbert modular
Eisenstein series. The second\, in collaboration with Yingkun Li and Jan V
onk\, considers the RM values of certain rigid meromorphic cocycles and p-
adic modular forms of weight 3/2 that arise as the fourier coefficients of
Zagier’s holomorphic kernel for the Shimura-Shintani correspondence.\n\
nFor an introductory lecture on this topic\, please see: https://www.youtu
be.com/watch?v=JsH1yMBwt_I .\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liyang Yang (Caltech)
DTSTART:20200715T170000Z
DTEND:20200715T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/8
DESCRIPTION:Title: Holomorphy of Adjoint $L$-functions for $\\GL(
n):$ $n\\leq 4$\nby Liyang Yang (Caltech) as part of Fields Number The
ory Seminar\n\n\nAbstract\nIn this talk\, we will mainly discuss holomorph
ic continuation of (complete) adjoint $L$-functions for $GL(n\,F)$ where
$n\\leq 4$ and $F$ is a number field. To obtain the continuation\, we gene
ralize Jacquet-Zagier's trace formula to $\\GL(n).$ Through this trace for
mula one can write adjoint L-functions as linear combinations of certain A
rtin $L$-series and $L$-functions defined by Langlands-Shahidi method and
Rankin-Selberg periods for non-discrete automorphic representations. A fur
ther application towards "Arithmetic Sato-Tate" for $GL(3)$ will be provid
ed as well.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Solomon Friedberg (Boston College)
DTSTART:20200805T170000Z
DTEND:20200805T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/9
DESCRIPTION:Title: Extending the theta correspondence\nby Sol
omon Friedberg (Boston College) as part of Fields Number Theory Seminar\n\
n\nAbstract\nThe classical theta correspondence establishes a relationship
between automorphic representations on special orthogonal groups and auto
morphic representations on symplectic groups or their double covers. This
correspondence is achieved by using as integral kernel a theta series that
is constructed from the Weil representation. In this talk I will briefly
survey earlier work on (local and global\, classical and other) theta corr
espondences and then present an extension of the classical theta correspon
dence to higher degree metaplectic covers. The key issue here is that for
higher degree covers there is no analogue of the Weil representation (or e
ven a minimal representation)\, so additional ingredients are needed. Join
t work with David Ginzburg.\n\nFor an introductory lecture on this topic\,
please see: https://youtu.be/p_jNha9u7DQ\n\nFor the introductory slides o
n this topic\, please see: http://www.fields.utoronto.ca/sites/default/fil
es/uploads/friedberg-background.pdf\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Payman Eskandari (University of Toronto)
DTSTART:20200812T170000Z
DTEND:20200812T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/10
DESCRIPTION:Title: Ceresa cycles of Fermat curves and Hodge theo
ry of fundamental groups\nby Payman Eskandari (University of Toronto)
as part of Fields Number Theory Seminar\n\n\nAbstract\nWe will show that t
he Ceresa cycles of Fermat curves of prime degree greater than 7 are of in
finite order modulo rational equivalence (i.e. in the Chow group). The pro
of combines several results due to B. Harris\, Pulte\, Kaenders and Darmon
-Rotger-Sols on the arithmetic and geometry of the mixed Hodge structure o
n the space of quadratic iterated integrals on an algebraic curve with a r
esult of Gross and Rohrlich on points of infinite order on Jacobians of Fe
rmat curves. This is a joint work with V. Kumar Murty.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clifton Cunningham (University of Calgary)
DTSTART:20200909T170000Z
DTEND:20200909T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/11
DESCRIPTION:Title: Arthur packets for unipotent representations
of the p-adic exceptional group G2\nby Clifton Cunningham (University
of Calgary) as part of Fields Number Theory Seminar\n\n\nAbstract\nAbstrac
t: This talk concerns work in progress on a generalization of the notion o
f local Arthur packets from Arthur-type representations of classical group
s over $p$-adic fields to all admissible representations of all connected
reductive algebraic groups over p-adic fields. In this talk our goal is mu
ch more modest: to report on this project for unipotent representations of
the exceptional group G_2(F) for a p-adic field F. We will explain how to
use the microlocal geometry of the moduli space of unramified Langlands p
arameters to compute what we call Adams-Barbasch-Vogan packets\, or ABV-pa
ckets for short\, for all unipotent representations of G_2(F) and how to f
ind the packet coefficients that are required to build stable distribution
s from ABV-packets. This talk will focus on the case that is the more inte
resting geometrically and will include a discussion of unipotent represent
ations that are not of Arthur type. We will argue that ABV-packets provide
the correct extension of the notion of Arthur packets by explaining that
the packet coefficients satisfy expected conditions coming from endoscopic
character identities.\n\nJoint work with Andrew Fiori and Qing Zhang\, ba
sed on earlier joint work with Andrew Fiori\, Ahmed Moussaoui\, James Mrac
ek and Bin Xu\, which in turn is based on earlier work by David Vogan.\n\n
For the introductory slides on this topic\, please see: http://www.fields.
utoronto.ca/sites/default/files/uploads/Introduction_0.pdf\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samit Dasgupta (Duke University)
DTSTART:20200826T160000Z
DTEND:20200826T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/12
DESCRIPTION:by Samit Dasgupta (Duke University) as part of Fields Number T
heory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ram Murty (Queen's University)
DTSTART:20200902T170000Z
DTEND:20200902T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/13
DESCRIPTION:Title: On the normal number of prime factors of sums
of Fourier coefficients of Hecke eigenforms\nby Ram Murty (Queen's Un
iversity) as part of Fields Number Theory Seminar\n\n\nAbstract\nIn 1984\,
Kumar Murty and I studied the normal number of prime factors of Fourier c
oefficients of modular forms. In recent joint work with Kumar Murty and Su
dhir Pujahari\, we study the normal number of prime factors of sums of Fou
rier coefficients of Hecke eigenforms using recent advances in the theory
of $\\ell$-adic Galois representations.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sinnou David (Université Pierre et Marie Curie)
DTSTART:20200916T170000Z
DTEND:20200916T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/14
DESCRIPTION:Title: On linear independence of special values of p
olylogarithms\nby Sinnou David (Université Pierre et Marie Curie) as
part of Fields Number Theory Seminar\n\n\nAbstract\nWe shall discuss a rec
ent joint work with Makoto Kawashima and Noriko Hirata-Kohno. For any set
of algebraic numbers in a fixed number field K satisfying standard metric
conditions in the theory (close enough to zero)\, we prove that the value
s of polylogarithms are linearly independent over K. This is done via a co
nstruction of an explicit system of Padé approximation. We shall also di
scuss extensions to lerch functions\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bennett (University of British Columbia)
DTSTART:20201014T170000Z
DTEND:20201014T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/15
DESCRIPTION:by Michael Bennett (University of British Columbia) as part of
Fields Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bennett (University of British Columbia)
DTSTART:20201102T170000Z
DTEND:20201102T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/17
DESCRIPTION:Title: Values of the Ramanujan tau-function\nby
Michael Bennett (University of British Columbia) as part of Fields Number
Theory Seminar\n\n\nAbstract\nIf a is an odd positive integer\, then a res
ult of Murty\, Murty and Shorey implies that there are at most finitely ma
ny positive integers n for which tau(n)=a\, where tau(n) is the Ramanujan
tau-function. In this talk\, I will discuss non-archimidean analogues of t
his result and show how the machinery of Frey curves and their associated
Galois representations can be employed to make such results explicit\, at
least in certain situations. Much of what I will discuss generalizes readi
ly to the more general situation of coefficients of cuspidal newforms of w
eight at least 4\, under natural arithmetic conditions. This is joint work
with Adela Gherga\, Vandita Patel and Samir Siksek.\n\nFor the introducto
ry slides on this topic\, please see: http://www.fields.utoronto.ca/sites/
default/files/uploads/Fields-2020.pdf\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debanjana Kundu (University of Toronto)
DTSTART:20201109T170000Z
DTEND:20201109T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/19
DESCRIPTION:Title: Iwasawa Theory of Fine Selmer Groups\nby
Debanjana Kundu (University of Toronto) as part of Fields Number Theory Se
minar\n\n\nAbstract\nInspired by the work of Iwasawa on growth of class gr
oups in Zp-extensions\, Mazur developed an analogous theory to study the g
rowth of Selmer groups of Abelian varieties in such extensions. In my pre-
talk\, I will introduce Iwasawa theory and briefly explain the work of Iwa
sawa and Mazur. I will finally introduce the fine Selmer group whose syste
matic study was initiated by Coates-Sujatha in 2005. This is a subgroup of
the Selmer group obtained by imposing stronger conditions at primes above
p.\n\nIn the main talk\, I will explain the growth of the fine Selmer gro
up in towers of number fields and relate it to the growth of class groups
in such towers. This will show the close relationship between the conjectu
res in classical Iwasawa theory and the Iwasawa theory of Abelian varietie
s. I will report on some modest progress made towards some of these conjec
tures. If time permits\, I will also talk about Control Theorems of Fine S
elmer Groups\n\nFor an introductory lecture on this topic\, please see: ht
tps://youtu.be/CiwR-YcEetI\n\nFor Introductory slides on this topic\, plea
se see: http://www.fields.utoronto.ca/sites/default/files/uploads/pre-talk
_0.pdf\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anup Dixit (Institute of Mathematical Sciences)
DTSTART:20201116T170000Z
DTEND:20201116T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/20
DESCRIPTION:Title: The generalised Diophantine m-tuples\nby
Anup Dixit (Institute of Mathematical Sciences) as part of Fields Number T
heory Seminar\n\n\nAbstract\nA set of natural numbers {a1\,a2\,⋯\,am} is
said to be a Diophantine m-tuple with property D(n) if aiaj+n is a perfec
t square for i≠j. One may ask\, what is the largest m for which such a t
uple exists. This problem has a long history\, attracting the attention of
many\, including Fermat\, Baker\, Davenport etc\, with significant progre
ss made in recent times due to Dujella and others. In this talk\, we consi
der a similar question by replacing the condition aiaj+n from being a squa
re to k-th powers. This is joint work with Ram Murty and Seoyoung Kim.\n\n
This talk will be accessible to graduate students!\n\nThe following link g
ives a quick introduction to the topic. It should help bring students up t
o speed on the history and recent developments on the problem:\n\nhttps://
web.math.pmf.unizg.hr/~duje/dtuples.html\n\nAdditional introductory readin
g material:\nhttp://www.fields.utoronto.ca/sites/default/files/uploads/Pal
ey%20graph_Diophantine%20tuples-RM_AG.pdf\nhttp://www.fields.utoronto.ca/s
ites/default/files/uploads/Diophantine%20tuples-RM_RB.pdf\nhttp://www.fiel
ds.utoronto.ca/sites/default/files/uploads/Diophantine%20tuples-Dujella.pd
f\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anantharam Raghuram (IISER Pune)
DTSTART:20201130T170000Z
DTEND:20201130T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/21
DESCRIPTION:Title: The arithmetic of Hecke characters and their
L-functions.\nby Anantharam Raghuram (IISER Pune) as part of Fields Nu
mber Theory Seminar\n\n\nAbstract\nI will begin by reviewing the notion of
an algebraic Hecke character over a number field in some depth. This part
of the talk should be accessible to a wide audience\; the only prerequisi
tes being basic algebraic number theory and some sheaf theory. The main go
al of the talk will be to discuss the critical values of L-functions attac
hed to algebraic Hecke characters. I will especially discuss certain signs
that appear in reciprocity laws satisfied by these special values. Toward
s the end\, I will talk about how this generalizes to L-functions for GL(n
) over a CM field.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Somnath Jha (Indian Institute of Technology\, Kanpur)
DTSTART:20201123T170000Z
DTEND:20201123T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/22
DESCRIPTION:Title: Algebraic functional equation for Selmer grou
ps\nby Somnath Jha (Indian Institute of Technology\, Kanpur) as part o
f Fields Number Theory Seminar\n\n\nAbstract\nSelmer groups of elliptic cu
rves encodes various aspects of the arithmetic of elliptic curves. In this
talk\, we will discuss a duality result for Selmer groups\, over a p-adic
Lie extension of a number field. This duality result can be thought of as
an algebraic functional equation.\n\nThis talk is based on joint works wi
th T. Ochiai and G. Zabradi\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanoli Gun (Institute of Mathematical Sciences\, Chennai)
DTSTART:20210118T170000Z
DTEND:20210118T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/23
DESCRIPTION:Title: On bounds of Fourier-coefficients of half-int
eger weight cusp forms\nby Sanoli Gun (Institute of Mathematical Scien
ces\, Chennai) as part of Fields Number Theory Seminar\n\n\nAbstract\nIn t
his talk\, we will discuss about omega results of Fourier-coefficients of
half-integer weight cusp forms which are not necessarily eigenforms. This
is a joint work with Kohnen and Soudararajan.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siddhi Pathak (The Pennsylvania State University)
DTSTART:20210125T170000Z
DTEND:20210125T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/24
DESCRIPTION:Title: The Okada space and vanishing of $L(1\,f)$\nby Siddhi Pathak (The Pennsylvania State University) as part of Fields
Number Theory Seminar\n\n\nAbstract\nFix a positive integer $N \\geq 2$. I
n this talk\, we will focus on the problem of determining all rational val
ued arithmetic functions\, periodic with period $N$ such that $L(1\,f) :=
\\sum_{n \\geq 1} f(n)/n = 0$. This study was initiated by S. Chowla in th
e 1960s\, drawing inspiration from Dirichlet's theorem that $L(1\,\\chi)\\
neq 0$ for a non-principal character $\\chi$. We will discuss recent joint
work with M. Ram Murty\, wherein we use a vanishing criterion of Okada to
construct an explicit basis for the $\\mathbb{Q}$-vector space of functio
ns $f \\pmod N$ such that $L(1\,f)=0$. This enables us to extend previous
works of Baker-Birch-Wirsing and Murty-Saradha\, with the arithmetic natur
e of Euler's constant $\\gamma$ emerging as an important theme.\n\nThis ta
lk will be accessible to graduate students.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asif Zaman (University of Toronto)
DTSTART:20210315T160000Z
DTEND:20210315T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/25
DESCRIPTION:Title: An approximate form of Artin's holomorphy con
jecture and nonvanishing of Artin L-functions\nby Asif Zaman (Universi
ty of Toronto) as part of Fields Number Theory Seminar\n\n\nAbstract\nLet
$k$ be a number field and $G$ be a finite group\, and let $\\mathfrak{F}_{
k}^{G}$ be a family of number fields $K$ such that $K/k$ is normal with Ga
lois group isomorphic to $G$. Together with Robert Lemke Oliver and Jess
e Thorner\, we prove for many families that for almost all $K \\in \\mathf
rak{F}_k^G$\, all of the $L$-functions associated to Artin representations
whose kernel does not contain a fixed normal subgroup are holomorphic and
non-vanishing in a wide region. \n\nThese results have several arithmetic
applications. For example\, we prove a strong effective prime ideal theor
em that holds for almost all fields in several natural large degree famil
ies\, including the family of degree $n$ $S_n$-extensions for any $n \\geq
2$ and the family of prime degree $p$ extensions (with any Galois structu
re) for any prime $p \\geq 2$. I will discuss this result\, describe the m
ain ideas of the proof\, and share some applications to bounds on $\\ell$-
torsion subgroups of class groups\, to the extremal order of class numbers
\, and to the subconvexity problem for Dedekind zeta functions.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenzhi Luo (Ohio State University)
DTSTART:20210301T170000Z
DTEND:20210301T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/26
DESCRIPTION:Title: Circle Method and Automorphic Forms\nby W
enzhi Luo (Ohio State University) as part of Fields Number Theory Seminar\
n\n\nAbstract\nIn this talk\, I will explain how to use the circle method
and the recently proved Vinogradov mean-value conjecture in the Waring's p
roblem\, to obtain power saving results for a class of shifted convolution
problems involving automorphic forms on GL(n).\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arul Shankar (University of Toronto)
DTSTART:20210201T170000Z
DTEND:20210201T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/27
DESCRIPTION:Title: Average sizes of the 2-torsion subgroups of t
he class groups in families of cubic fields\nby Arul Shankar (Universi
ty of Toronto) as part of Fields Number Theory Seminar\n\n\nAbstract\nThe
Cohen--Lenstra--Martinet conjectures have been verified in only two cases.
Davenport--Heilbronn compute the average size of the 3-torsion subgroups
in the class group of quadratic fields and Bhargava computes the average s
ize of the 2-torsion subgroups in the class groups of cubic fields. The va
lues computed in the above two results are very stable. In particular\, wo
rk of Bhargava--Varma shows that they do not change if one instead average
s over the family of quadratic or cubic fields satisfying any finite set o
f splitting conditions.\n\nHowever for certain "thin" families of cubic fi
elds\, namely\, families of monogenic and n-monogenic cubic fields\, the s
tory is very different. In this talk\, we will determine the average size
of the 2-torsion subgroups of the class groups of fields in these thin fam
ilies. Surprisingly\, these values differ from the Cohen--Lenstra--Martine
t predictions! We will also provide an explanation for this difference in
terms of the Tamagawa numbers of naturally arising reductive groups. This
is joint work with Manjul Bhargava and Jon Hanke.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Carmen Cojocaru (University of Illinois at Chicago)
DTSTART:20210208T170000Z
DTEND:20210208T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/28
DESCRIPTION:Title: Bounds for the distribution of Frobenius trac
es associated to products of non-CM elliptic curves\nby Alina Carmen C
ojocaru (University of Illinois at Chicago) as part of Fields Number Theor
y Seminar\n\n\nAbstract\nLet $E_1/\\Q\, \\ldots\, E_g/\\Q$ be elliptic cur
ves over $\\Q$\, without complex multiplication and pairwise non-isogenous
over $\\overline{\\Q}$. For an integer $t$ and a positive real number $x$
\, denote by $\\pi_A(x\, t)$ the number of primes $p \\leq x$\, of good re
duction for the abelian variety $A := E_1 \\times \\ldots \\times E_g$\, f
or which the Frobenius trace associated to the reduction of $A$ modulo $p$
equals $t$. We present unconditional and conditional upper bounds for $\\
pi_A(x\, t)$. This is joint work with Tian Wang (University of Illinois at
Chicago).\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meng Fai Lim (Central China Normal University)
DTSTART:20210222T170000Z
DTEND:20210222T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/29
DESCRIPTION:Title: On etale wild kernel and a conjecture of Gree
nberg\nby Meng Fai Lim (Central China Normal University) as part of Fi
elds Number Theory Seminar\n\n\nAbstract\nIn this talk\, we shall study th
e growth of the etale wild kernels in various p-adic Lie extensions. The e
tale wild kernels (coined by Banaszak\, Kolster\, Nguyen Quand Do etc) are
related to the special values of the Dedekind zeta function. In this talk
\, we reinterpret the etale wild kernel as an appropriate fine Selmer grou
p in the sense of Coates-Sujatha. This viewpoint brings us to the problem
of studying a control theorem of the said fine Selmer groups\, which in tu
rn allows us to minic the strategies developed by Greenberg. However\, thi
s improvisation is not a direct procedure\, as one needs to estimate the g
rowth of cohomology groups of open subgroups of p-adic Lie groups which is
not accessible directly from the lie algebraic approach of Greenberg (one
of the main issue is that the open subgroups have the same lie algebra an
d so the cohomology of the lie algebra cannot distinguish the cohomology g
roups of the subgroups). Among the tools used in estimatiing these cohomol
ogy groups\, one notable ingredient is Tate's lemma which asserts the vani
shing of the first Γ\n\n-cohomology groups of nonzero Tate twist of Qp/Zp
.\n\nOnce we have such a control theorem\, we apply them to obtain asympto
tic growth formulas for the etale wild kernels in various said p-adic Lie
extensions. The leading terms of the growth formulas are related to a cert
ain Galois group\, and an appropriate noncommutative variant of Greenberg'
s conjecture predicts that this said Galois group is not "too big". In par
ticular\, we shall see that Greenberg conjecture gives an asymptotic upper
bound on the growth of the etale wild kernels. These upper bound are not
necessarily always optimal. Indeed\, building on calculations of Sharifi\,
we can give some examples which show that the etale wild kernels can grow
much slower than the predicted estimate of Greenberg.\n\nFinally\, if tim
e permits\, we shall mention briefly on a joint work with Debanajana Kundu
on the fine Selmer groups of elliptic curves which builds on a natural an
alogue/generalization of Tate's lemma in the elliptic situation.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Akbary (University of Lethbridge)
DTSTART:20210308T170000Z
DTEND:20210308T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/30
DESCRIPTION:Title: Sums of triangular numbers and sums of square
s\nby Amir Akbary (University of Lethbridge) as part of Fields Number
Theory Seminar\n\n\nAbstract\nFor non-negative integers $a$\, $b$\, and $n
$\, let $t(a\,b\;n)$ be the number of representations of $n$ as a sum of t
riangular numbers with coefficients $1$ or $3$ and let $r(a\, b\; n)$ be t
he number of representations of $n$ as a sum of squares with coefficients
$1$ or $3$. It is known that for $a$ and $b$ satisfying $1\\leq a+3b \\leq
7$\, we have $$ t(a\,b\;{n}) = \\frac{2}{2+{a\\choose4}+ab} r(a\,b\;8n+a+
3b) $$ and for $a$ and $b$ satisfying $a+3b=8$\, we have $$ t(a\,b\;{n}) =
\\frac{2}{2+{a\\choose4}+ab} \\left( r(a\,b\;8n+a+3b) - r(a\,b\; (8n+a+3b
)/4) \\right). $$ Such identities are not known for $a+3b>8$. \n\nWe repor
t on our joint work with Zafer Selcuk Aygin (University of Calgary) in whi
ch\, for general $a$ and $b$ with $a+b$ even\, we prove asymptotic equival
ence of formulas similar to the above\, as $n\\rightarrow\\infty$. One of
our main results extends a theorem of Bateman\, Datskovsky\, and Knopp whe
re the case $b=0$ and general $a$ was considered. Our approach is differen
t from Bateman-Datskovsky-Knopp's proof where the circle method and singul
ar series were used. We achieve our results by explicitly computing the Ei
senstein components of the generating functions of $t(a\,b\;n)$ and $r(a\,
b\;n)$.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeff Achter (Colorado State University)
DTSTART:20210412T160000Z
DTEND:20210412T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/31
DESCRIPTION:Title: Arithmetic occult periods\nby Jeff Achter
(Colorado State University) as part of Fields Number Theory Seminar\n\n\n
Abstract\nSeveral natural complex configuration spaces admit surprising un
iformizations as arithmetic ball quotients\, by identifying each parametri
zed object with the periods of some auxiliary object. In each case\, the t
heory of canonical models of Shimura varieties gives the ball quotient the
structure of a variety over the ring of integers of a cyclotomic field.
I will show that these (transcendentally-defined) period maps respect thes
e algebraic structures\, and thus are actually arithmetic.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leo Goldmakher (Williams College)
DTSTART:20210322T160000Z
DTEND:20210322T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/32
DESCRIPTION:Title: Khovanskii's theorem and effective results on
sumset structure\nby Leo Goldmakher (Williams College) as part of Fie
lds Number Theory Seminar\n\n\nAbstract\nA remarkable theorem due to Khova
nskii asserts that for any finite subset A of an abelian semigroup\, the c
ardinality of the h-fold sumset hA grows like a polynomial for all suffici
ently large h. However\, neither the polynomial nor what sufficiently larg
e means are understood in general. In joint work with Michael Curran (Oxfo
rd)\, we obtain an effective version of Khovanskii's theorem for any subse
t of $\\mathbb{Z}^d$ whose convex hull is a simplex\; previously such resu
lts were only available for d=1. Our approach also gives information about
the structure of hA\, answering a recent question posed by Granville and
Shakan. The talk will be broadly accessible\; interested mathematicians fr
om any field are encouraged to attend.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dinakar Ramakrishnan (California Institute of Technology)
DTSTART:20210405T160000Z
DTEND:20210405T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/33
DESCRIPTION:Title: Central L-values of U(3) x U(2)\, non-vanishi
ng and subconvexity\nby Dinakar Ramakrishnan (California Institute of
Technology) as part of Fields Number Theory Seminar\n\n\nAbstract\nThis is
a report on joint work with Philippe Michel and Liyang Yang. Let f\, resp
. g\, denote a holomorphic cusp form on U(2\,1)\, resp. U(1\,1)\, of weigh
t (-k\, k/2)\, resp. k\, for an even integer k \, taken to be at least 260
for technical reasons. We consider the Rankin-Selberg L-function L(s\, f
x g)\, which is defined by base changing to GL over the relevant imaginary
quadratic field. We average over a family of f with varying square-free l
evels\, and establish a non-vanishing result at the central value\, as wel
l as a hybrid subconvexity in the level aspect.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Payman Eskandari (University of Toronto)
DTSTART:20210510T160000Z
DTEND:20210510T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/34
DESCRIPTION:Title: Extension classes of a mixed motive and subgr
oups of the unipotent radical of the motivic Galois group\nby Payman E
skandari (University of Toronto) as part of Fields Number Theory Seminar\n
\n\nAbstract\nThe fundamental difference between pure motives (roughly spe
aking coming from cohomology of smooth projective varieties) and mixed mot
ives (coming from cohomology of arbitrary varieties) is existence of nontr
ivial extensions in the latter setting. The unipotent radical of the motiv
ic Galois group of a mixed motive M is intimately related to the extension
data in the category generated by M. A fairly recent result of Deligne de
scribes this unipotent radical in terms of the extensions 0⟶WpM⟶M⟶M/
WpM⟶0 collectively (W⋅ being the weight filtration). We shall recall t
his result and then discuss what information each of these extensions indi
vidually contains. We will end the talk with an application to motives who
se unipotent radical of the motivic Galois group is as large as possible\,
and discuss some examples in the category of mixed Tate motives over Q. T
his is a joint work with Kumar Murty.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Armin Jamshidpey (University of Waterloo)
DTSTART:20210517T160000Z
DTEND:20210517T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/35
DESCRIPTION:Title: A survey on quantum algorithms for computing
class groups\nby Armin Jamshidpey (University of Waterloo) as part of
Fields Number Theory Seminar\n\n\nAbstract\nOur goal is to review the exis
ting quantum algorithms to compute class groups of number fields. First we
briefly survey results for this problem on a standard “classical” com
puter. After a short introduction to quantum computing\, we look at the hi
dden subgroup problem and quantum algorithms for it\, as a fundamental too
l. Finally we present the polynomial-time quantum algorithm for computing
the ideal class group (under the Generalized Riemann Hypothesis) introduce
d by Biasse and Song (2016).\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Jossen (ETH Zürich)
DTSTART:20210614T160000Z
DTEND:20210614T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/36
DESCRIPTION:Title: On Exponential motives and their Fundamental
Groups\nby Peter Jossen (ETH Zürich) as part of Fields Number Theory
Seminar\n\n\nAbstract\nI will construct several cohomology theories for pa
irs (X\,f) consisting of an algebraic variety X and a regular function f o
n X\, and explain how to produce a universal cohomology theory for such pa
irs. This universal cohomology theory takes its values in the tannakian ca
tegory of Exponential Motives. I will then show by means of concrete examp
les how to compute some interesting tannakian fundamental groups of expone
ntial motives\, and explain their relevance for transcendence questions.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristiana Bertolin (University of Torino)
DTSTART:20210621T160000Z
DTEND:20210621T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/37
DESCRIPTION:Title: Grothendieck period conjecture and 1-motives<
/a>\nby Cristiana Bertolin (University of Torino) as part of Fields Number
Theory Seminar\n\n\nAbstract\nWe start computing the periods and the dime
nsion of the motivic Galois group of 1-motives. Then we apply Grothendieck
period conjecture to 1-motives and we will see some consequences of this
conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART:20210628T160000Z
DTEND:20210628T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/38
DESCRIPTION:Title: Arithmetic Siegel-Weil formula for GSpin Shim
ura varieties\nby Chao Li (Columbia University) as part of Fields Numb
er Theory Seminar\n\n\nAbstract\nWe prove a semi-global arithmetic Siegel-
Weil formula as conjectured by Kudla\, which relates the arithmetic inters
ection numbers of special cycles on GSpin Shimura varieties at a place of
good reduction and the central derivatives of nonsingular Fourier coeffici
ents of Siegel Eisenstein series. We will motivate this conjecture and dis
cuss some aspects of the proof.\n\nThis is joint work with Wei Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Hardouin (Institut de Mathématiques de Toulouse)
DTSTART:20210719T160000Z
DTEND:20210719T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/40
DESCRIPTION:Title: Elliptic surfaces and the enumeration of wal
ks with small steps in the quarter plane\nby Charlotte Hardouin (Insti
tut de Mathématiques de Toulouse) as part of Fields Number Theory Seminar
\n\n\nAbstract\nA walk in the quarter plane is a path between integral poi
nts of the plane that uses a prescribed set of directions and remains in
the first quadrant. In the past years\, the enumeration of such walks has
attracted the attentionof many authors in combinatorics and probability.
The complexity of their enumeration is encoded in the algebraic nature of
their associated generatingseries. The main questions are: are these serie
s algebraic\, holonomic (solutions of linear differential equations) or di
fferentially algebraic (solutions of algebraicdifferential equations)? In
this talk\, we will show how this algebraic nature can be understood via t
he study of a discrete functional equation over a curve E of genus zero or
one over a function field . In the genus zero case\, the functional equat
ion corresponds to a socalled q-difference equation and the generating ser
ies is always differentially transcendental. In genus one\, the dynamic of
the functional equation is the addition by agiven point P of the elliptic
curve E.In that situation\, the nature of the generating series is enti
rely captured by the linear dependence relations of certain prescribed po
ints in the Mordell-Weil lattice of the elliptic surface attached to E.
This are joint works with T. Dreyfus\, J. Roques and M.F. Singer.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sean Howe (University of Utah)
DTSTART:20210712T160000Z
DTEND:20210712T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/42
DESCRIPTION:Title: A tale of two analyticities\nby Sean Howe
(University of Utah) as part of Fields Number Theory Seminar\n\n\nAbstrac
t\nIf X/S is a family of smooth projective varieties over \\overline{Q}\,
then the Hodge filtration on the cohomology of X induces locally an analyt
ic period map from S(C) to a flag variety. The flag variety also has a nat
ural algebraic structure over \\overline{Q}\, and it is natural to ask: wh
ich (\\overline{Q}-)algebraic conditions on the Hodge filtration induce (\
\overline{Q}-)algebraic conditions on S? \nIf the Hodge conjecture holds\,
then the condition that a given rational cohomology class on a tensor pow
er of the cohomology be a Hodge cycle\, which is evidently algebraic on th
e flag variety\, is also \\overline{Q}-algebraic on S (if we only ask that
it be C-algebraic then this is a theorem of Cattani-Deligne-Kaplan). More
over\, if the Grothendieck Period Conjecture also holds\, then using a res
ult of Andre on the existence of an \\overline{Q}-rational Hodge generic p
oint it can be shown that these are essentially the only ones. For familie
s of abelian varieties\, there is an unconditional proof of this bialgebra
icity theorem due to Ullmo-Yafaev. \nIn joint work in progress with Christ
ian Klevdal\, we investigate a local p-adic analytic analog of this story:
now X/S is a smooth proper family of rigid analytic varieties defined ove
r a p-adic field\, and we ask when rigid analytic conditions on the Hodge-
Tate filtration on p-adic etale cohomology induce rigid analytic condition
s on S. In this talk I will explain some of our results\, with an emphasis
on the connection between our strategy of proof and the conjectural strat
egy described above over Q.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anwesh Ray (University of British Columbia)
DTSTART:20210809T160000Z
DTEND:20210809T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/43
DESCRIPTION:Title: Arithmetic statistics and the Iwasawa theory
of elliptic curves\nby Anwesh Ray (University of British Columbia) as
part of Fields Number Theory Seminar\n\n\nAbstract\nAn elliptic curve defi
ned over the rationals gives rise to a compatible system of Galois represe
ntations. The Iwasawa invariants associated to these representations epito
mize their arithmetic and Iwasawa theoretic properties. The study of these
invariants is the subject of much conjecture and contemplation. For insta
nce\, according to a long-standing conjecture of R. Greenberg\, the Iwasaw
a "mu-invariant" must vanish\, subject to mild hypothesis. Overall\, there
is a subtle relationship between the behavior of these invariants and the
p-adic Birch and Swinnerton-Dyer formula. We study the behaviour of these
invariants on average\, where elliptic curves over the rationals are orde
red according to height. I will discuss recent results joint with Debanjan
a Kundu\, in which we set out new directions in arithmetic statistics and
Iwasawa theory.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Herbert Gangl (Durham University)
DTSTART:20210726T160000Z
DTEND:20210726T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/44
DESCRIPTION:Title: Zagier's Polylogarithm Conjecture revisited\nby Herbert Gangl (Durham University) as part of Fields Number Theory S
eminar\n\n\nAbstract\nInstigated by work of Borel and Bloch\, Zagier formu
lated his Polylogarithm Conjecture in the late eighties and proved it for
weight 2. After a flurry of activity and advances at the time\, notably by
Goncharov who not only provided powerful new tools for a proof in weight
3 but also set out a vast program with a plethora of conjectural statement
s for attacking it\, progress seemed to be stalled for a number of years.
More recently\, a solution to one of Goncharov's central conjectures in we
ight 4 has been given. Moreover\, by adopting a new point of view\, work b
y Goncharov and Rudenko gave a proof of the original conjecture in weight
4. In this impressionist talk I intend to give a rough idea of the develop
ments from the early days on\, avoiding most of the technical bits\, and a
lso hint at a number of recent results for higher weight (joint with S.Cha
rlton and D.Radchenko).\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Michel (École Polytechnique Fédérale de Lausanne)
DTSTART:20210913T160000Z
DTEND:20210913T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/45
DESCRIPTION:Title: Algebraic Twists of automorphic L-functions\nby Philippe Michel (École Polytechnique Fédérale de Lausanne) as pa
rt of Fields Number Theory Seminar\n\n\nAbstract\nLet $L(\\pi\,s)=\\sum_{n
\\geq 1}\\lambda(n)/n^{s}$ be an automorphic $L$-function. \n\nFor $q$ a p
rime number and $\\chi(q)$ a non-trivial multiplicative character\, the $\
\chi$ twisted $L$-function is (essentially) given by \n\n$$L(\\pi.\\chi\,s
)=\\sum_{n\\geq 1}\\lambda(n)\\chi(n)/n^{s}.$$\n\nThe subconvexity problem
(in the $\\chi$-aspect) aims at bounding non-trivially $L(\\pi.\\chi\,s)$
when $\\Re s=1/2$ and has now been resolved in a number of cases.\n\nIn t
his talk\, we discuss a series of works joint with E. Fouvry\, E. Kowalski
\, Y. Lin and W. Sawin regarding a generalisation of this problem when $\\
chi$ is replaced by a more general function\n\n$$K:{\\mathbb Z}/q :{\\math
bb Z}\\rightarrow {\\mathbb C}$$\n\nand $L(\\pi.\\chi\,s)$ is replaced by
the the $K$ algebraically twisted $L$-series\n\n$$L(\\pi.K\,s)=\\sum_{n\\g
eq 1}\\lambda(n)K(n)/n^{s}.$$\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhishek Oswal (The California Institute of Technology)
DTSTART:20210927T160000Z
DTEND:20210927T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/46
DESCRIPTION:by Abhishek Oswal (The California Institute of Technology) as
part of Fields Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Lubotzky (Hebrew University of Jerusalem)
DTSTART:20210920T160000Z
DTEND:20210920T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/47
DESCRIPTION:Title: From Ramanujan graphs to Ramanujan complexes<
/a>\nby Alexander Lubotzky (Hebrew University of Jerusalem) as part of Fie
lds Number Theory Seminar\n\n\nAbstract\nRamanujan graphs are k-regular gr
aphs with all nontrivial eigenvalues bounded (in absolute value) by $2\\sq
rt{k-1}$. They are optimal expanders (from a spectral point of view). Expl
icit constructions of such graphs were given in the 80's as quotients of t
he Bruhat-Tits tree associated with $GL(2)$ over a local field $F$\, by th
e action of suitable congruence subgroups of arithmetic groups.\n\nThe spe
ctral bound was proved using the works of Hecke\, Deligne and Drinfeld on
the"Ramanujan conjecture" in the theory of automorphic forms.\n\nThe work
of Lafforgue\, extending Drinfeld from $GL(2)$ to $GL(n)$\, opened the doo
r for the construction of Ramanujan complexes as quotients of the Bruhat-T
its buildings associated with $GL(n)$ over $F$.\n\nThis way one gets finit
e simplicial complexes\, which on one hand are "random like ''and at the s
ame time have strong symmetries. These seemingly contradicting properties
make them very useful for constructions of various external objects.\n\nVa
rious applications have been found in combinatorics\, coding theory and in
relation to Gromov's overlapping properties.\n\nWe will survey some of th
ese applications.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clément Dupont (Université de Montpellier)
DTSTART:20211004T160000Z
DTEND:20211004T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/48
DESCRIPTION:Title: Constructing extensions in mixed Tate motives
\nby Clément Dupont (Université de Montpellier) as part of Fields Nu
mber Theory Seminar\n\n\nAbstract\nThe category of mixed Tate motives (ite
rated extensions of the pure Tate motives Q(-n) for all integers n) is wel
l understood from an abstract point of view but the structure of its perio
ds is still very mysterious. This includes open questions on special value
s of Dedekind zeta functions\, and irrationality questions for higher regu
lators. It is therefore an important task to understand certain extensions
in mixed Tate motives as explicitly and geometrically as possible. In thi
s talk I will survey different aspects of this problem\, and explain in pa
rticular a construction of « small » motives whose periods are odd zeta
values.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jayan Mukherjee (Brown University)
DTSTART:20211018T160000Z
DTEND:20211018T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/49
DESCRIPTION:Title: Tautological families of stacks of cyclic cov
ers of projective spaces\nby Jayan Mukherjee (Brown University) as par
t of Fields Number Theory Seminar\n\n\nAbstract\nIn this article\, we stud
y the existence of tautological families on a Zariski open set of the coar
se moduli space parametrizing certain Galois covers over projective spaces
. More specifically\, let ($1$) $\\mathscr{H}_{n.r.d}$ (resp. $M_{n\,r\,d}
$) be the stack (resp. coarse moduli) parametrizing smooth simple cyclic c
overs of degree $r$ over the projective space $\\mathbb{P}^n$ branched alo
ng a divisor of degree $rd$\, and ($2$) $\\mathscr{H}_{1\,3\,d_1\,d_2}$ (r
esp. $M_{1\,3\,d_1\,d_2}$) be the stack (resp. coarse moduli) of smooth cy
clic triple covers over $\\mathbb{P}^1$. In the former case\, we show that
such a family exists if and only if $\\textrm{gcd}(rd\, n+1) \\mid d$ whi
le in the latter case we show that it always exists. We further show that
even when such a family exists\, often it cannot be extended to the open l
ocus of objects without extra automorphisms. The existence of tautological
families on a Zariski open set of its coarse moduli can be interpreted in
terms of rationality of the stack if the coarse moduli space is rational.
Combining our results with known results on the rationality of the coarse
moduli we obtain results on rationality of the above stacks when $n=1$ or
$n=2$. Our study is motivated by the study of stacks of hyperelliptic cur
ves by Gorchinskiy and Viviani.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Klingler (Humboldt University of Berlin)
DTSTART:20211129T170000Z
DTEND:20211129T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/50
DESCRIPTION:by Bruno Klingler (Humboldt University of Berlin) as part of F
ields Number Theory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kumar Murty (The Fields Institute and University of Toronto)
DTSTART:20211101T160000Z
DTEND:20211101T170000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/52
DESCRIPTION:Title: Non-vanishing of Poincare series\nby Kuma
r Murty (The Fields Institute and University of Toronto) as part of Fields
Number Theory Seminar\n\n\nAbstract\nWe consider an estimate of Rankin on
the number of non-zero Poincare series for the full modular group and ind
icate how it can be improved.\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Stubley (McGill University)
DTSTART:20211115T170000Z
DTEND:20211115T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/53
DESCRIPTION:by Eric Stubley (McGill University) as part of Fields Number T
heory Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yves André (Institut de Mathématiques de Jussieu-Paris Rive Gauc
he and Sorbonne Université)
DTSTART:20211206T170000Z
DTEND:20211206T180000Z
DTSTAMP:20260404T095031Z
UID:fields-number-theory-seminar/54
DESCRIPTION:by Yves André (Institut de Mathématiques de Jussieu-Paris Ri
ve Gauche and Sorbonne Université) as part of Fields Number Theory Semina
r\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/fields-number-theory-sem
inar/54/
END:VEVENT
END:VCALENDAR