BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Michael Harris (Columbia University)
DTSTART;VALUE=DATE-TIME:20200513T170000Z
DTEND;VALUE=DATE-TIME:20200513T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/1
DESCRIPTION:Title: Local Langlands parametrization for G2\nby Michae
l Harris (Columbia University) as part of Fields Number Theory Seminar\n\n
\nAbstract\nPlease register for this talk here: https://zoom.us/meeting/re
gister/tJwlduurrzMqE9PidEG3TfLJ2qHG9dEp5cua . \n\nThis is a report on join
t work with C. Khare and J. Thorne. We construct a Langlands parametrizati
on of supercuspidal representations of $G_2$ over a $p$-adic fields. More
precisely\, for any finite extension $K/\\mathbb{Q}_p$ we will construct a
bijection \n\n$\\mathcal{L}_g : \\mathcal{A}_g^0 (G_2\, K)\\rightarrow \\
mathcal{G}^0(G_2\, K)$\n\nfrom the set of generic supercuspidal representa
tions of $G_2(K)$ to the set of irreducible continuous homomorphisms $\\rh
o: W_K \\rightarrow G_2(\\mathbb{C})$ with $W_K$ the Weil group of $K$. Th
e construction of the map is simply a matter of assembling arguments that
are already in literature\, plus an unpublished result of Savin (included
as an appendix in our article) on the global genericity of an exceptional
theta lift. The proof of surjectivity is an application of a recent result
of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$. This all
ows us to carry out a strategy\, based on automorphy lifting theorems\, th
at was initially developed in our joint work with G. Böckle on potential
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://researchseminars.org/talk/fields-number-theory-seminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritabrata Munshi (Tata Institute of Fundamental Research and India
n Statistical Institute)
DTSTART;VALUE=DATE-TIME:20200520T170000Z
DTEND;VALUE=DATE-TIME:20200520T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/2
DESCRIPTION:Title: Circle method and subconvexity\nby Ritabrata Muns
hi (Tata Institute of Fundamental Research and Indian Statistical Institut
e) as part of Fields Number Theory Seminar\n\n\nAbstract\nPlease register
for this talk here: https://zoom.us/meeting/register/tJwlduurrzMqE9PidEG3T
fLJ2qHG9dEp5cua .\n\nIn this talk I will discuss the subconvexity problem
for GL(3)xGL(2) Rankin-Selberg L-functions. I will show how the delta meth
od can be used to prove various subconvexity bounds for such L-functions.
The talk should be accessible to graduate students.\n\nFor the introductor
y lecture for this talk\, please see: https://youtu.be/LJeyes0BpOk .\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sujatha Ramdorai (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20200527T170000Z
DTEND;VALUE=DATE-TIME:20200527T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/3
DESCRIPTION:Title: Euler characteristics and Arithmetic\nby Sujatha
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 Euler char
acteristic of the Selmer groups of elliptic curves encodes information on
the arithmetic of the elliptic curve. We discuss the connection to the Bir
ch and Swinnerton Dyer conjecture and study the Euler characteristics of r
esidually isomorphic Galois representations.\n\nFor the introductory lectu
re to this talk\, please see: https://youtu.be/YYh9TodNPdQ .\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chandrashekhar Khare (UCLA)
DTSTART;VALUE=DATE-TIME:20200603T170000Z
DTEND;VALUE=DATE-TIME:20200603T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/4
DESCRIPTION:Title: Wiles defect for Hecke algebras that are not complete
intersections\nby Chandrashekhar Khare (UCLA) as part of Fields Numbe
r Theory Seminar\n\n\nAbstract\nPlease register for this talk here: https:
//zoom.us/meeting/register/tJwlduurrzMqE9PidEG3TfLJ2qHG9dEp5cua .\n\nIn hi
s work on modularity theorems\, Wiles proved a numerical criterion for a m
ap of rings R->T to be an isomorphism of complete intersections. In additi
on to proving modularity theorems\, this numerical criterion also implies
a connection between the order of a certain Selmer group and a special val
ue of an L-function.\n\nIn this talk I will consider the case of a Hecke a
lgebra acting on the cohomology a Shimura curve associated to a quaternion
algebra. In this case\, one has an analogous map of rings R->T which is k
nown to be an isomorphism\, but in many cases the rings R and T fail to be
complete intersections. This means that Wiles's numerical criterion will
fail to hold.\n\nI will describe a method for precisely computing the exte
nt to which the numerical criterion fails (i.e. the 'Wiles defect") at a n
ewform f which gives rise to an augmentation T -> Z_p. The defect turns ou
t to be determined entirely by local information at the primes q dividing
the discriminant of the quaternion algebra at which the mod p representati
on arising from f is ``trivial''. (For instance if f corresponds to a semi
stable elliptic curve\, then the local defect at q is related to the ``tam
e regulator'' of the Tate period of the elliptic curve at q.)\n\nFurther r
eading: \n\nMy talk will be based on:\n\nWiles defect for Hecke algebras t
hat are not complete intersection\njoint with G. Böckle and Jeff Manning\
navailable at: https://arxiv.org/abs/1910.08507\n\nAnother reference to gi
ve some context:\n\nAUTHOR = Ribet\, Kenneth A\nTITLE = Multiplicities of
Galois representations in Jacobians of Shimura curves\nBOOKTITLE = Festsch
rift in honor of I.I. Piatetski-Shapiro on the occasion of his sixtieth bi
rthday\, Part II Ramat Aviv\, 1989\nSERIES = Israel Math. Conf. Proc.\nVOL
UME = 3\nPAGES = 221--236\nPUBLISHER = Weizmann\, Jerusalem\nYEAR = 1990\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mahesh Kakde (Indian Institute of Science)
DTSTART;VALUE=DATE-TIME:20200610T170000Z
DTEND;VALUE=DATE-TIME:20200610T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/5
DESCRIPTION:Title: Integral Gross-Stark conjecture and explicit formulae
for Brumer-Stark units\nby Mahesh Kakde (Indian Institute of Science)
as part of Fields Number Theory Seminar\n\n\nAbstract\nPlease register fo
r this talk here: https://zoom.us/meeting/register/tJwlduurrzMqE9PidEG3TfL
J2qHG9dEp5cua .\n\nIn this talk I will present integral refinements of the
Gross-Stark conjecture\, due to Gross\, known as the tower of fields conj
ecture. I will then describe my on-going work proving this conjecture. The
tower of fields conjecture implies a conjecture of Dasgupta that gives an
explicit p-adic analytic formula for the Brumer-Stark units. I will sketc
h this along with an application to Hilbert’s 12th problem. This is all
a joint work with Samit Dasgupta.\n\nFor the introductory lecture to this
talk\, please see: https://youtu.be/oJIfdh1ibH4\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen Kudla (University of Toronto)
DTSTART;VALUE=DATE-TIME:20200617T170000Z
DTEND;VALUE=DATE-TIME:20200617T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/6
DESCRIPTION:Title: On the subring of special cycles on orthogonal Shimur
a varieties\nby Stephen Kudla (University of Toronto) as part of Field
s Number Theory Seminar\n\n\nAbstract\nPlease register for this talk here:
https://zoom.us/meeting/register/tJwlduurrzMqE9PidEG3TfLJ2qHG9dEp5cua .\n
\nPlease refer to the introductory slides for this talk here: http://www.f
ields.utoronto.ca/sites/default/files/uploads/Fields.prep.talk.2020.pdf .
\n\nBy old results with Millson\, the generating series for the cohomology
classes of special cycles on orthogonal Shimura varieties over a totally
real field are Hilbert-Siegel modular forms. These forms arise via theta s
eries. Using this result and the Siegel-Weil formula\, we show that the pr
oducts in the subring of cohomology generated by the special cycles are co
ntrolled by the Fourier coefficients of triple pullbacks of certain Siegel
-Eisenstein series. As a consequence\, there are comparison isomorphisms b
etween special subrings for different Shimura varieties that may be of mot
ivic origin.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henri Darmon (McGill University)
DTSTART;VALUE=DATE-TIME:20200708T170000Z
DTEND;VALUE=DATE-TIME:20200708T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/7
DESCRIPTION:Title: Generating series for RM values of rigid meromorphic
cocycles\nby Henri Darmon (McGill University) as part of Fields Number
Theory Seminar\n\n\nAbstract\nPlease register here: https://zoom.us/meeti
ng/register/tJEtcuuvqTksHdfsLV9ZiYeWaHDdqnotA5-5 .\n\nI will describe two
ongoing works whose unifying theme is to establish the algebraicity of the
RM values of rigid meromorphic cocycles\, by realizing these invariants a
s the fourier coefficients of certain p-adic modular generating series\, t
hereby obtaining the desired properties from the theory of deformations of
Galois representations and from global class field theory. The first proj
ect\, in collaboration with Alice Pozzi and Jan Vonk\, considers the RM va
lues of the Dedekind-Rademacher cocycle and its relation to the diagonal r
estrictions of certain first order deformations of Hilbert modular Eisenst
ein series. The second\, in collaboration with Yingkun Li and Jan Vonk\, c
onsiders the RM values of certain rigid meromorphic cocycles and p-adic mo
dular 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.youtube.com/
watch?v=JsH1yMBwt_I .\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liyang Yang (Caltech)
DTSTART;VALUE=DATE-TIME:20200715T170000Z
DTEND;VALUE=DATE-TIME:20200715T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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 Theory Sem
inar\n\n\nAbstract\nIn this talk\, we will mainly discuss holomorphic cont
inuation of (complete) adjoint $L$-functions for $GL(n\,F)$ where $n\\leq
4$ and $F$ is a number field. To obtain the continuation\, we generalize
Jacquet-Zagier's trace formula to $\\GL(n).$ Through this trace formula on
e can write adjoint L-functions as linear combinations of certain Artin $L
$-series and $L$-functions defined by Langlands-Shahidi method and Rankin-
Selberg periods for non-discrete automorphic representations. A further ap
plication towards "Arithmetic Sato-Tate" for $GL(3)$ will be provided as w
ell.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Solomon Friedberg (Boston College)
DTSTART;VALUE=DATE-TIME:20200805T170000Z
DTEND;VALUE=DATE-TIME:20200805T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/9
DESCRIPTION:Title: Extending the theta correspondence\nby Solomon Fr
iedberg (Boston College) as part of Fields Number Theory Seminar\n\n\nAbst
ract\nThe classical theta correspondence establishes a relationship betwee
n automorphic representations on special orthogonal groups and automorphic
representations on symplectic groups or their double covers. This corresp
ondence is achieved by using as integral kernel a theta series that is con
structed from the Weil representation. In this talk I will briefly survey
earlier work on (local and global\, classical and other) theta corresponde
nces and then present an extension of the classical theta correspondence t
o higher degree metaplectic covers. The key issue here is that for higher
degree covers there is no analogue of the Weil representation (or even a m
inimal representation)\, so additional ingredients are needed. Joint work
with David Ginzburg.\n\nFor an introductory lecture on this topic\, please
see: https://youtu.be/p_jNha9u7DQ\n\nFor the introductory slides on this
topic\, please see: http://www.fields.utoronto.ca/sites/default/files/uplo
ads/friedberg-background.pdf\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Payman Eskandari (University of Toronto)
DTSTART;VALUE=DATE-TIME:20200812T170000Z
DTEND;VALUE=DATE-TIME:20200812T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/10
DESCRIPTION:Title: Ceresa cycles of Fermat curves and Hodge theory of f
undamental groups\nby Payman Eskandari (University of Toronto) as part
of Fields Number Theory Seminar\n\n\nAbstract\nWe will show that the Cere
sa cycles of Fermat curves of prime degree greater than 7 are of infinite
order modulo rational equivalence (i.e. in the Chow group). The proof comb
ines several results due to B. Harris\, Pulte\, Kaenders and Darmon-Rotger
-Sols on the arithmetic and geometry of the mixed Hodge structure on the s
pace of quadratic iterated integrals on an algebraic curve with a result o
f Gross and Rohrlich on points of infinite order on Jacobians of Fermat cu
rves. This is a joint work with V. Kumar Murty.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/10
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clifton Cunningham (University of Calgary)
DTSTART;VALUE=DATE-TIME:20200909T170000Z
DTEND;VALUE=DATE-TIME:20200909T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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 Calg
ary) as part of Fields Number Theory Seminar\n\n\nAbstract\nAbstract: This
talk concerns work in progress on a generalization of the notion of local
Arthur packets from Arthur-type representations of classical groups over
$p$-adic fields to all admissible representations of all connected reducti
ve algebraic groups over p-adic fields. In this talk our goal is much more
modest: to report on this project for unipotent representations of the ex
ceptional group G_2(F) for a p-adic field F. We will explain how to use th
e microlocal geometry of the moduli space of unramified Langlands paramete
rs to compute what we call Adams-Barbasch-Vogan packets\, or ABV-packets f
or short\, for all unipotent representations of G_2(F) and how to find the
packet coefficients that are required to build stable distributions from
ABV-packets. This talk will focus on the case that is the more interesting
geometrically and will include a discussion of unipotent representations
that are not of Arthur type. We will argue that ABV-packets provide the co
rrect extension of the notion of Arthur packets by explaining that the pac
ket coefficients satisfy expected conditions coming from endoscopic charac
ter identities.\n\nJoint work with Andrew Fiori and Qing Zhang\, based on
earlier joint work with Andrew Fiori\, Ahmed Moussaoui\, James Mracek and
Bin Xu\, which in turn is based on earlier work by David Vogan.\n\nFor the
introductory slides on this topic\, please see: http://www.fields.utoront
o.ca/sites/default/files/uploads/Introduction_0.pdf\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/11
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samit Dasgupta (Duke University)
DTSTART;VALUE=DATE-TIME:20200826T160000Z
DTEND;VALUE=DATE-TIME:20200826T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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://researchseminars.org/talk/fields-number-theory-seminar/12
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ram Murty (Queen's University)
DTSTART;VALUE=DATE-TIME:20200902T170000Z
DTEND;VALUE=DATE-TIME:20200902T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/13
DESCRIPTION:Title: On the normal number of prime factors of sums of Fou
rier coefficients of Hecke eigenforms\nby Ram Murty (Queen's Universit
y) 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 coeffici
ents of modular forms. In recent joint work with Kumar Murty and Sudhir Pu
jahari\, we study the normal number of prime factors of sums of Fourier co
efficients of Hecke eigenforms using recent advances in the theory of $\\e
ll$-adic Galois representations.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/13
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sinnou David (Université Pierre et Marie Curie)
DTSTART;VALUE=DATE-TIME:20200916T170000Z
DTEND;VALUE=DATE-TIME:20200916T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/14
DESCRIPTION:Title: On linear independence of special values of polyloga
rithms\nby Sinnou David (Université Pierre et Marie Curie) as part of
Fields Number Theory Seminar\n\n\nAbstract\nWe shall discuss a recent joi
nt work with Makoto Kawashima and Noriko Hirata-Kohno. For any set of alge
braic numbers in a fixed number field K satisfying standard metric conditi
ons in the theory (close enough to zero)\, we prove that the values of po
lylogarithms are linearly independent over K. This is done via a construct
ion of an explicit system of Padé approximation. We shall also discuss e
xtensions to lerch functions\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/14
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bennett (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20201014T170000Z
DTEND;VALUE=DATE-TIME:20201014T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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://researchseminars.org/talk/fields-number-theory-seminar/15
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bennett (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20201102T170000Z
DTEND;VALUE=DATE-TIME:20201102T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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 result of
Murty\, Murty and Shorey implies that there are at most finitely many posi
tive integers n for which tau(n)=a\, where tau(n) is the Ramanujan tau-fun
ction. In this talk\, I will discuss non-archimidean analogues of this res
ult and show how the machinery of Frey curves and their associated Galois
representations can be employed to make such results explicit\, at least i
n certain situations. Much of what I will discuss generalizes readily to t
he more general situation of coefficients of cuspidal newforms of weight a
t least 4\, under natural arithmetic conditions. This is joint work with A
dela Gherga\, Vandita Patel and Samir Siksek.\n\nFor the introductory slid
es on this topic\, please see: http://www.fields.utoronto.ca/sites/default
/files/uploads/Fields-2020.pdf\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/17
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debanjana Kundu (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201109T170000Z
DTEND;VALUE=DATE-TIME:20201109T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/19
DESCRIPTION:Title: Iwasawa Theory of Fine Selmer Groups\nby Debanja
na Kundu (University of Toronto) as part of Fields Number Theory Seminar\n
\n\nAbstract\nInspired by the work of Iwasawa on growth of class groups in
Zp-extensions\, Mazur developed an analogous theory to study the growth o
f Selmer groups of Abelian varieties in such extensions. In my pre-talk\,
I will introduce Iwasawa theory and briefly explain the work of Iwasawa an
d Mazur. I will finally introduce the fine Selmer group whose systematic s
tudy was initiated by Coates-Sujatha in 2005. This is a subgroup of the Se
lmer group obtained by imposing stronger conditions at primes above p.\n\n
In the main talk\, I will explain the growth of the fine Selmer group in t
owers of number fields and relate it to the growth of class groups in such
towers. This will show the close relationship between the conjectures in
classical Iwasawa theory and the Iwasawa theory of Abelian varieties. I wi
ll report on some modest progress made towards some of these conjectures.
If time permits\, I will also talk about Control Theorems of Fine Selmer G
roups\n\nFor an introductory lecture on this topic\, please see: https://y
outu.be/CiwR-YcEetI\n\nFor Introductory slides on this topic\, please see:
http://www.fields.utoronto.ca/sites/default/files/uploads/pre-talk_0.pdf\
n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/19
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anup Dixit (Institute of Mathematical Sciences)
DTSTART;VALUE=DATE-TIME:20201116T170000Z
DTEND;VALUE=DATE-TIME:20201116T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/20
DESCRIPTION:Title: The generalised Diophantine m-tuples\nby Anup Di
xit (Institute of Mathematical Sciences) as part of Fields Number Theory S
eminar\n\n\nAbstract\nA set of natural numbers {a1\,a2\,⋯\,am} is said t
o be a Diophantine m-tuple with property D(n) if aiaj+n is a perfect squar
e for i≠j. One may ask\, what is the largest m for which such a tuple ex
ists. This problem has a long history\, attracting the attention of many\,
including Fermat\, Baker\, Davenport etc\, with significant progress made
in recent times due to Dujella and others. In this talk\, we consider a s
imilar question by replacing the condition aiaj+n from being a square to k
-th powers. This is joint work with Ram Murty and Seoyoung Kim.\n\nThis ta
lk will be accessible to graduate students!\n\nThe following link gives a
quick introduction to the topic. It should help bring students up to speed
on the history and recent developments on the problem:\n\nhttps://web.mat
h.pmf.unizg.hr/~duje/dtuples.html\n\nAdditional introductory reading mater
ial:\nhttp://www.fields.utoronto.ca/sites/default/files/uploads/Paley%20gr
aph_Diophantine%20tuples-RM_AG.pdf\nhttp://www.fields.utoronto.ca/sites/de
fault/files/uploads/Diophantine%20tuples-RM_RB.pdf\nhttp://www.fields.utor
onto.ca/sites/default/files/uploads/Diophantine%20tuples-Dujella.pdf\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/20
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anantharam Raghuram (IISER Pune)
DTSTART;VALUE=DATE-TIME:20201130T170000Z
DTEND;VALUE=DATE-TIME:20201130T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/21
DESCRIPTION:Title: The arithmetic of Hecke characters and their L-funct
ions.\nby Anantharam Raghuram (IISER Pune) as part of Fields Number Th
eory Seminar\n\n\nAbstract\nI will begin by reviewing the notion of an alg
ebraic Hecke character over a number field in some depth. This part of the
talk should be accessible to a wide audience\; the only prerequisites bei
ng basic algebraic number theory and some sheaf theory. The main goal of t
he talk will be to discuss the critical values of L-functions attached to
algebraic Hecke characters. I will especially discuss certain signs that a
ppear in reciprocity laws satisfied by these special values. Towards the e
nd\, I will talk about how this generalizes to L-functions for GL(n) over
a CM field.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/21
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Somnath Jha (Indian Institute of Technology\, Kanpur)
DTSTART;VALUE=DATE-TIME:20201123T170000Z
DTEND;VALUE=DATE-TIME:20201123T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/22
DESCRIPTION:Title: Algebraic functional equation for Selmer groups\
nby Somnath Jha (Indian Institute of Technology\, Kanpur) as part of Field
s Number Theory Seminar\n\n\nAbstract\nSelmer groups of elliptic curves en
codes 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 ex
tension of a number field. This duality result can be thought of as an alg
ebraic functional equation.\n\nThis talk is based on joint works with T. O
chiai and G. Zabradi\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/22
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanoli Gun (Institute of Mathematical Sciences\, Chennai)
DTSTART;VALUE=DATE-TIME:20210118T170000Z
DTEND;VALUE=DATE-TIME:20210118T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/23
DESCRIPTION:Title: On bounds of Fourier-coefficients of half-integer we
ight cusp forms\nby Sanoli Gun (Institute of Mathematical Sciences\, C
hennai) as part of Fields Number Theory Seminar\n\n\nAbstract\nIn this tal
k\, we will discuss about omega results of Fourier-coefficients of half-in
teger weight cusp forms which are not necessarily eigenforms. This is a jo
int work with Kohnen and Soudararajan.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/23
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siddhi Pathak (The Pennsylvania State University)
DTSTART;VALUE=DATE-TIME:20210125T170000Z
DTEND;VALUE=DATE-TIME:20210125T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/24
DESCRIPTION:Title: The Okada space and vanishing of $L(1\,f)$\nby S
iddhi Pathak (The Pennsylvania State University) as part of Fields Number
Theory Seminar\n\n\nAbstract\nFix a positive integer $N \\geq 2$. In this
talk\, we will focus on the problem of determining all rational valued ari
thmetic 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 the 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 w
ith M. Ram Murty\, wherein we use a vanishing criterion of Okada to constr
uct an explicit basis for the $\\mathbb{Q}$-vector space of functions $f \
\pmod N$ such that $L(1\,f)=0$. This enables us to extend previous works o
f Baker-Birch-Wirsing and Murty-Saradha\, with the arithmetic nature of Eu
ler's constant $\\gamma$ emerging as an important theme.\n\nThis talk will
be accessible to graduate students.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/24
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asif Zaman (University of Toronto)
DTSTART;VALUE=DATE-TIME:20210315T160000Z
DTEND;VALUE=DATE-TIME:20210315T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/25
DESCRIPTION:Title: An approximate form of Artin's holomorphy conjecture
and nonvanishing of Artin L-functions\nby Asif Zaman (University of T
oronto) 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 Galois gr
oup isomorphic to $G$. Together with Robert Lemke Oliver and Jesse Thorn
er\, we prove for many families that for almost all $K \\in \\mathfrak{F}_
k^G$\, all of the $L$-functions associated to Artin representations whose
kernel does not contain a fixed normal subgroup are holomorphic and non-va
nishing in a wide region. \n\nThese results have several arithmetic applic
ations. For example\, we prove a strong effective prime ideal theorem tha
t holds for almost all fields in several natural large degree families\, i
ncluding the family of degree $n$ $S_n$-extensions for any $n \\geq 2$ and
the family of prime degree $p$ extensions (with any Galois structure) for
any prime $p \\geq 2$. I will discuss this result\, describe the main ide
as 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://researchseminars.org/talk/fields-number-theory-seminar/25
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenzhi Luo (Ohio State University)
DTSTART;VALUE=DATE-TIME:20210301T170000Z
DTEND;VALUE=DATE-TIME:20210301T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/26
DESCRIPTION:Title: Circle Method and Automorphic Forms\nby Wenzhi L
uo (Ohio State University) as part of Fields Number Theory Seminar\n\n\nAb
stract\nIn this talk\, I will explain how to use the circle method and the
recently proved Vinogradov mean-value conjecture in the Waring's problem\
, to obtain power saving results for a class of shifted convolution proble
ms involving automorphic forms on GL(n).\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/26
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arul Shankar (University of Toronto)
DTSTART;VALUE=DATE-TIME:20210201T170000Z
DTEND;VALUE=DATE-TIME:20210201T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/27
DESCRIPTION:Title: Average sizes of the 2-torsion subgroups of the clas
s groups in families of cubic fields\nby Arul Shankar (University of T
oronto) as part of Fields Number Theory Seminar\n\n\nAbstract\nThe Cohen--
Lenstra--Martinet conjectures have been verified in only two cases. Davenp
ort--Heilbronn compute the average size of the 3-torsion subgroups in the
class group of quadratic fields and Bhargava computes the average size of
the 2-torsion subgroups in the class groups of cubic fields. The values co
mputed in the above two results are very stable. In particular\, work of B
hargava--Varma shows that they do not change if one instead averages over
the family of quadratic or cubic fields satisfying any finite set of split
ting conditions.\n\nHowever for certain "thin" families of cubic fields\,
namely\, families of monogenic and n-monogenic cubic fields\, the story 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 families.
Surprisingly\, these values differ from the Cohen--Lenstra--Martinet predi
ctions! We will also provide an explanation for this difference in terms o
f the Tamagawa numbers of naturally arising reductive groups. This is join
t work with Manjul Bhargava and Jon Hanke.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/27
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Carmen Cojocaru (University of Illinois at Chicago)
DTSTART;VALUE=DATE-TIME:20210208T170000Z
DTEND;VALUE=DATE-TIME:20210208T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/28
DESCRIPTION:Title: Bounds for the distribution of Frobenius traces asso
ciated to products of non-CM elliptic curves\nby Alina Carmen Cojocaru
(University of Illinois at Chicago) as part of Fields Number Theory Semin
ar\n\n\nAbstract\nLet $E_1/\\Q\, \\ldots\, E_g/\\Q$ be elliptic curves ove
r $\\Q$\, without complex multiplication and pairwise non-isogenous over $
\\overline{\\Q}$. For an integer $t$ and a positive real number $x$\, deno
te by $\\pi_A(x\, t)$ the number of primes $p \\leq x$\, of good reduction
for the abelian variety $A := E_1 \\times \\ldots \\times E_g$\, for whic
h 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 Chicag
o).\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/28
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meng Fai Lim (Central China Normal University)
DTSTART;VALUE=DATE-TIME:20210222T170000Z
DTEND;VALUE=DATE-TIME:20210222T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/29
DESCRIPTION:Title: On etale wild kernel and a conjecture of Greenberg\nby Meng Fai Lim (Central China Normal University) as part of Fields Nu
mber Theory Seminar\n\n\nAbstract\nIn this talk\, we shall study the growt
h of the etale wild kernels in various p-adic Lie extensions. The etale wi
ld kernels (coined by Banaszak\, Kolster\, Nguyen Quand Do etc) are relate
d to the special values of the Dedekind zeta function. In this talk\, we r
einterpret the etale wild kernel as an appropriate fine Selmer group in th
e sense of Coates-Sujatha. This viewpoint brings us to the problem of stud
ying a control theorem of the said fine Selmer groups\, which in turn allo
ws us to minic the strategies developed by Greenberg. However\, this impro
visation is not a direct procedure\, as one needs to estimate the growth o
f cohomology groups of open subgroups of p-adic Lie groups which is not ac
cessible directly from the lie algebraic approach of Greenberg (one of the
main issue is that the open subgroups have the same lie algebra and so th
e cohomology of the lie algebra cannot distinguish the cohomology groups o
f the subgroups). Among the tools used in estimatiing these cohomology gro
ups\, one notable ingredient is Tate's lemma which asserts the vanishing o
f the first Γ\n\n-cohomology groups of nonzero Tate twist of Qp/Zp.\n\nOn
ce we have such a control theorem\, we apply them to obtain asymptotic gro
wth formulas for the etale wild kernels in various said p-adic Lie extensi
ons. The leading terms of the growth formulas are related to a certain Gal
ois group\, and an appropriate noncommutative variant of Greenberg's conje
cture predicts that this said Galois group is not "too big". In particular
\, we shall see that Greenberg conjecture gives an asymptotic upper bound
on the growth of the etale wild kernels. These upper bound are not necessa
rily always optimal. Indeed\, building on calculations of Sharifi\, we can
give some examples which show that the etale wild kernels can grow much s
lower than the predicted estimate of Greenberg.\n\nFinally\, if time permi
ts\, we shall mention briefly on a joint work with Debanajana Kundu on the
fine Selmer groups of elliptic curves which builds on a natural analogue/
generalization of Tate's lemma in the elliptic situation.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/29
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Akbary (University of Lethbridge)
DTSTART;VALUE=DATE-TIME:20210308T170000Z
DTEND;VALUE=DATE-TIME:20210308T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/30
DESCRIPTION:Title: Sums of triangular numbers and sums of squares\n
by 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 triangul
ar numbers with coefficients $1$ or $3$ and let $r(a\, b\; n)$ be the numb
er of representations of $n$ as a sum of squares with coefficients $1$ or
$3$. It is known that for $a$ and $b$ satisfying $1\\leq a+3b \\leq 7$\, w
e 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 report on ou
r joint work with Zafer Selcuk Aygin (University of Calgary) in which\, fo
r general $a$ and $b$ with $a+b$ even\, we prove asymptotic equivalence of
formulas similar to the above\, as $n\\rightarrow\\infty$. One of our mai
n results extends a theorem of Bateman\, Datskovsky\, and Knopp where the
case $b=0$ and general $a$ was considered. Our approach is different from
Bateman-Datskovsky-Knopp's proof where the circle method and singular seri
es were used. We achieve our results by explicitly computing the Eisenstei
n components of the generating functions of $t(a\,b\;n)$ and $r(a\,b\;n)$.
\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/30
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeff Achter (Colorado State University)
DTSTART;VALUE=DATE-TIME:20210412T160000Z
DTEND;VALUE=DATE-TIME:20210412T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/31
DESCRIPTION:Title: Arithmetic occult periods\nby Jeff Achter (Color
ado State University) as part of Fields Number Theory Seminar\n\n\nAbstrac
t\nSeveral natural complex configuration spaces admit surprising uniformiz
ations as arithmetic ball quotients\, by identifying each parametrized obj
ect with the periods of some auxiliary object. In each case\, the theory o
f canonical models of Shimura varieties gives the ball quotient the struct
ure of a variety over the ring of integers of a cyclotomic field. I will
show that these (transcendentally-defined) period maps respect these algeb
raic structures\, and thus are actually arithmetic.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/31
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leo Goldmakher (Williams College)
DTSTART;VALUE=DATE-TIME:20210322T160000Z
DTEND;VALUE=DATE-TIME:20210322T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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 Fields Num
ber Theory Seminar\n\n\nAbstract\nA remarkable theorem due to Khovanskii a
sserts that for any finite subset A of an abelian semigroup\, the cardinal
ity of the h-fold sumset hA grows like a polynomial for all sufficiently l
arge h. However\, neither the polynomial nor what sufficiently large means
are understood in general. In joint work with Michael Curran (Oxford)\, w
e obtain an effective version of Khovanskii's theorem for any subset of $\
\mathbb{Z}^d$ whose convex hull is a simplex\; previously such results wer
e only available for d=1. Our approach also gives information about the st
ructure of hA\, answering a recent question posed by Granville and Shakan.
The talk will be broadly accessible\; interested mathematicians from any
field are encouraged to attend.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/32
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dinakar Ramakrishnan (California Institute of Technology)
DTSTART;VALUE=DATE-TIME:20210405T160000Z
DTEND;VALUE=DATE-TIME:20210405T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/33
DESCRIPTION:Title: Central L-values of U(3) x U(2)\, non-vanishing and
subconvexity\nby Dinakar Ramakrishnan (California Institute of Technol
ogy) as part of Fields Number Theory Seminar\n\n\nAbstract\nThis is a repo
rt on joint work with Philippe Michel and Liyang Yang. Let f\, resp. g\, d
enote a holomorphic cusp form on U(2\,1)\, resp. U(1\,1)\, of weight (-k\,
k/2)\, resp. k\, for an even integer k \, taken to be at least 260 for te
chnical 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 quadra
tic field. We average over a family of f with varying square-free levels\,
and establish a non-vanishing result at the central value\, as well as a
hybrid subconvexity in the level aspect.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/33
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Payman Eskandari (University of Toronto)
DTSTART;VALUE=DATE-TIME:20210510T160000Z
DTEND;VALUE=DATE-TIME:20210510T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/34
DESCRIPTION:Title: Extension classes of a mixed motive and subgroups of
the unipotent radical of the motivic Galois group\nby Payman Eskandar
i (University of Toronto) as part of Fields Number Theory Seminar\n\n\nAbs
tract\nThe fundamental difference between pure motives (roughly speaking c
oming from cohomology of smooth projective varieties) and mixed motives (c
oming from cohomology of arbitrary varieties) is existence of nontrivial e
xtensions in the latter setting. The unipotent radical of the motivic Galo
is group of a mixed motive M is intimately related to the extension data i
n the category generated by M. A fairly recent result of Deligne describes
this unipotent radical in terms of the extensions 0⟶WpM⟶M⟶M/WpM⟶0
collectively (W⋅ being the weight filtration). We shall recall this res
ult and then discuss what information each of these extensions individuall
y contains. We will end the talk with an application to motives whose unip
otent radical of the motivic Galois group is as large as possible\, and di
scuss some examples in the category of mixed Tate motives over Q. This is
a joint work with Kumar Murty.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/34
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Armin Jamshidpey (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20210517T160000Z
DTEND;VALUE=DATE-TIME:20210517T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/35
DESCRIPTION:Title: A survey on quantum algorithms for computing class g
roups\nby Armin Jamshidpey (University of Waterloo) as part of Fields
Number Theory Seminar\n\n\nAbstract\nOur goal is to review the existing qu
antum algorithms to compute class groups of number fields. First we briefl
y survey results for this problem on a standard “classical” computer.
After a short introduction to quantum computing\, we look at the hidden su
bgroup problem and quantum algorithms for it\, as a fundamental tool. Fina
lly we present the polynomial-time quantum algorithm for computing the ide
al class group (under the Generalized Riemann Hypothesis) introduced by Bi
asse and Song (2016).\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/35
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Jossen (ETH Zürich)
DTSTART;VALUE=DATE-TIME:20210614T160000Z
DTEND;VALUE=DATE-TIME:20210614T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/36
DESCRIPTION:Title: On Exponential motives and their Fundamental Groups<
/a>\nby Peter Jossen (ETH Zürich) as part of Fields Number Theory Seminar
\n\n\nAbstract\nI will construct several cohomology theories for pairs (X\
,f) consisting of an algebraic variety X and a regular function f on X\, a
nd explain how to produce a universal cohomology theory for such pairs. Th
is universal cohomology theory takes its values in the tannakian category
of Exponential Motives. I will then show by means of concrete examples how
to compute some interesting tannakian fundamental groups of exponential m
otives\, and explain their relevance for transcendence questions.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/36
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristiana Bertolin (University of Torino)
DTSTART;VALUE=DATE-TIME:20210621T160000Z
DTEND;VALUE=DATE-TIME:20210621T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/37
DESCRIPTION:Title: Grothendieck period conjecture and 1-motives\nby
Cristiana Bertolin (University of Torino) as part of Fields Number Theory
Seminar\n\n\nAbstract\nWe start computing the periods and the dimension o
f the motivic Galois group of 1-motives. Then we apply Grothendieck period
conjecture to 1-motives and we will see some consequences of this conject
ure.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/37
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART;VALUE=DATE-TIME:20210628T160000Z
DTEND;VALUE=DATE-TIME:20210628T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/38
DESCRIPTION:Title: Arithmetic Siegel-Weil formula for GSpin Shimura var
ieties\nby Chao Li (Columbia University) as part of Fields Number Theo
ry Seminar\n\n\nAbstract\nWe prove a semi-global arithmetic Siegel-Weil fo
rmula as conjectured by Kudla\, which relates the arithmetic intersection
numbers of special cycles on GSpin Shimura varieties at a place of good re
duction and the central derivatives of nonsingular Fourier coefficients of
Siegel Eisenstein series. We will motivate this conjecture and discuss so
me aspects of the proof.\n\nThis is joint work with Wei Zhang.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/38
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Hardouin (Institut de Mathématiques de Toulouse)
DTSTART;VALUE=DATE-TIME:20210719T160000Z
DTEND;VALUE=DATE-TIME:20210719T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/40
DESCRIPTION:Title: Elliptic surfaces and the enumeration of walks with
small steps in the quarter plane\nby Charlotte Hardouin (Institut de
Mathématiques de Toulouse) as part of Fields Number Theory Seminar\n\n\nA
bstract\nA walk in the quarter plane is a path between integral points of
the plane that uses a prescribed set of directions and remains in the fir
st quadrant. In the past years\, the enumeration of such walks has attrac
ted the attentionof many authors in combinatorics and probability. The com
plexity of their enumeration is encoded in the algebraic nature of their a
ssociated generatingseries. The main questions are: are these series algeb
raic\, holonomic (solutions of linear differential equations) or different
ially algebraic (solutions of algebraicdifferential equations)? In this ta
lk\, we will show how this algebraic nature can be understood via the stud
y of a discrete functional equation over a curve E of genus zero or one ov
er a function field . In the genus zero case\, the functional equation cor
responds to a socalled q-difference equation and the generating series is
always differentially transcendental. In genus one\, the dynamic of the fu
nctional equation is the addition by agiven point P of the elliptic curve
E.In that situation\, the nature of the generating series is entirely c
aptured by the linear dependence relations of certain prescribed points in
the Mordell-Weil lattice of the elliptic surface attached to E. This a
re joint works with T. Dreyfus\, J. Roques and M.F. Singer.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/40
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sean Howe (University of Utah)
DTSTART;VALUE=DATE-TIME:20210712T160000Z
DTEND;VALUE=DATE-TIME:20210712T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/42
DESCRIPTION:Title: A tale of two analyticities\nby Sean Howe (Unive
rsity of Utah) as part of Fields Number Theory Seminar\n\n\nAbstract\nIf X
/S is a family of smooth projective varieties over \\overline{Q}\, then th
e Hodge filtration on the cohomology of X induces locally an analytic peri
od map from S(C) to a flag variety. The flag variety also has a natural al
gebraic structure over \\overline{Q}\, and it is natural to ask: which (\\
overline{Q}-)algebraic conditions on the Hodge filtration induce (\\overli
ne{Q}-)algebraic conditions on S? \nIf the Hodge conjecture holds\, then t
he condition that a given rational cohomology class on a tensor power of t
he cohomology be a Hodge cycle\, which is evidently algebraic on the 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). Moreover\,
if the Grothendieck Period Conjecture also holds\, then using a result of
Andre on the existence of an \\overline{Q}-rational Hodge generic point it
can be shown that these are essentially the only ones. For families of ab
elian varieties\, there is an unconditional proof of this bialgebraicity t
heorem due to Ullmo-Yafaev. \nIn joint work in progress with Christian Kle
vdal\, we investigate a local p-adic analytic analog of this story: now X/
S is a smooth proper family of rigid analytic varieties defined over a p-a
dic field\, and we ask when rigid analytic conditions on the Hodge-Tate fi
ltration on p-adic etale cohomology induce rigid analytic conditions 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 strategy des
cribed above over Q.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/42
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anwesh Ray (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20210809T160000Z
DTEND;VALUE=DATE-TIME:20210809T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/43
DESCRIPTION:Title: Arithmetic statistics and the Iwasawa theory of elli
ptic curves\nby Anwesh Ray (University of British Columbia) as part of
Fields Number Theory Seminar\n\n\nAbstract\nAn elliptic curve defined ove
r the rationals gives rise to a compatible system of Galois representation
s. The Iwasawa invariants associated to these representations epitomize th
eir arithmetic and Iwasawa theoretic properties. The study of these invari
ants is the subject of much conjecture and contemplation. For instance\, a
ccording to a long-standing conjecture of R. Greenberg\, the Iwasawa "mu-i
nvariant" must vanish\, subject to mild hypothesis. Overall\, there is a s
ubtle relationship between the behavior of these invariants and the p-adic
Birch and Swinnerton-Dyer formula. We study the behaviour of these invari
ants on average\, where elliptic curves over the rationals are ordered acc
ording to height. I will discuss recent results joint with Debanjana Kundu
\, in which we set out new directions in arithmetic statistics and Iwasawa
theory.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/43
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Herbert Gangl (Durham University)
DTSTART;VALUE=DATE-TIME:20210726T160000Z
DTEND;VALUE=DATE-TIME:20210726T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/44
DESCRIPTION:Title: Zagier's Polylogarithm Conjecture revisited\nby
Herbert Gangl (Durham University) as part of Fields Number Theory Seminar\
n\n\nAbstract\nInstigated by work of Borel and Bloch\, Zagier formulated h
is Polylogarithm Conjecture in the late eighties and proved it for weight
2. After a flurry of activity and advances at the time\, notably by Goncha
rov who not only provided powerful new tools for a proof in weight 3 but a
lso set out a vast program with a plethora of conjectural statements for a
ttacking it\, progress seemed to be stalled for a number of years. More re
cently\, a solution to one of Goncharov's central conjectures in weight 4
has been given. Moreover\, by adopting a new point of view\, work by Gonch
arov and Rudenko gave a proof of the original conjecture in weight 4. In t
his impressionist talk I intend to give a rough idea of the developments f
rom the early days on\, avoiding most of the technical bits\, and also hin
t at a number of recent results for higher weight (joint with S.Charlton a
nd D.Radchenko).\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/44
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippe Michel (École Polytechnique Fédérale de Lausanne)
DTSTART;VALUE=DATE-TIME:20210913T160000Z
DTEND;VALUE=DATE-TIME:20210913T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/45
DESCRIPTION:Title: Algebraic Twists of automorphic L-functions\nby
Philippe Michel (École Polytechnique Fédérale de Lausanne) as part of F
ields 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 prime nu
mber and $\\chi(q)$ a non-trivial multiplicative character\, the $\\chi$ t
wisted $L$-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 th
e $\\chi$-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 this tal
k\, we discuss a series of works joint with E. Fouvry\, E. Kowalski\, Y. L
in and W. Sawin regarding a generalisation of this problem when $\\chi$ is
replaced by a more general function\n\n$$K:{\\mathbb Z}/q :{\\mathbb Z}\\
rightarrow {\\mathbb C}$$\n\nand $L(\\pi.\\chi\,s)$ is replaced by the the
$K$ algebraically twisted $L$-series\n\n$$L(\\pi.K\,s)=\\sum_{n\\geq 1}\\
lambda(n)K(n)/n^{s}.$$\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/45
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhishek Oswal (The California Institute of Technology)
DTSTART;VALUE=DATE-TIME:20210927T160000Z
DTEND;VALUE=DATE-TIME:20210927T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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://researchseminars.org/talk/fields-number-theory-seminar/46
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Lubotzky (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20210920T160000Z
DTEND;VALUE=DATE-TIME:20210920T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/47
DESCRIPTION:Title: From Ramanujan graphs to Ramanujan complexes\nby
Alexander Lubotzky (Hebrew University of Jerusalem) as part of Fields Num
ber Theory Seminar\n\n\nAbstract\nRamanujan graphs are k-regular graphs wi
th all nontrivial eigenvalues bounded (in absolute value) by $2\\sqrt{k-1}
$. They are optimal expanders (from a spectral point of view). Explicit co
nstructions of such graphs were given in the 80's as quotients of the Bruh
at-Tits tree associated with $GL(2)$ over a local field $F$\, by the actio
n of suitable congruence subgroups of arithmetic groups.\n\nThe spectral b
ound was proved using the works of Hecke\, Deligne and Drinfeld on the"Ram
anujan conjecture" in the theory of automorphic forms.\n\nThe work of Laff
orgue\, extending Drinfeld from $GL(2)$ to $GL(n)$\, opened the door for t
he construction of Ramanujan complexes as quotients of the Bruhat-Tits bui
ldings associated with $GL(n)$ over $F$.\n\nThis way one gets finite simpl
icial complexes\, which on one hand are "random like ''and at the same tim
e have strong symmetries. These seemingly contradicting properties make th
em very useful for constructions of various external objects.\n\nVarious a
pplications have been found in combinatorics\, coding theory and in relati
on to Gromov's overlapping properties.\n\nWe will survey some of these app
lications.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/47
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clément Dupont (Université de Montpellier)
DTSTART;VALUE=DATE-TIME:20211004T160000Z
DTEND;VALUE=DATE-TIME:20211004T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/48
DESCRIPTION:Title: Constructing extensions in mixed Tate motives\nb
y Clément Dupont (Université de Montpellier) as part of Fields Number Th
eory Seminar\n\n\nAbstract\nThe category of mixed Tate motives (iterated e
xtensions of the pure Tate motives Q(-n) for all integers n) is well under
stood from an abstract point of view but the structure of its periods is s
till very mysterious. This includes open questions on special values of De
dekind zeta functions\, and irrationality questions for higher regulators.
It is therefore an important task to understand certain extensions in mix
ed Tate motives as explicitly and geometrically as possible. In this talk
I will survey different aspects of this problem\, and explain in particula
r a construction of « small » motives whose periods are odd zeta values.
\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/48
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jayan Mukherjee (Brown University)
DTSTART;VALUE=DATE-TIME:20211018T160000Z
DTEND;VALUE=DATE-TIME:20211018T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/49
DESCRIPTION:Title: Tautological families of stacks of cyclic covers of
projective spaces\nby Jayan Mukherjee (Brown University) as part of Fi
elds Number Theory Seminar\n\n\nAbstract\nIn this article\, we study the e
xistence of tautological families on a Zariski open set of the coarse modu
li space parametrizing certain Galois covers over projective spaces. More
specifically\, let ($1$) $\\mathscr{H}_{n.r.d}$ (resp. $M_{n\,r\,d}$) be t
he stack (resp. coarse moduli) parametrizing smooth simple cyclic covers o
f degree $r$ over the projective space $\\mathbb{P}^n$ branched along a di
visor of degree $rd$\, and ($2$) $\\mathscr{H}_{1\,3\,d_1\,d_2}$ (resp. $M
_{1\,3\,d_1\,d_2}$) be the stack (resp. coarse moduli) of smooth cyclic tr
iple covers over $\\mathbb{P}^1$. In the former case\, we show that such a
family exists if and only if $\\textrm{gcd}(rd\, n+1) \\mid d$ while in t
he latter case we show that it always exists. We further show that even wh
en such a family exists\, often it cannot be extended to the open locus of
objects without extra automorphisms. The existence of tautological famili
es 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. Combin
ing 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 curves by
Gorchinskiy and Viviani.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/49
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Klingler (Humboldt University of Berlin)
DTSTART;VALUE=DATE-TIME:20211129T170000Z
DTEND;VALUE=DATE-TIME:20211129T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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://researchseminars.org/talk/fields-number-theory-seminar/50
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kumar Murty (The Fields Institute and University of Toronto)
DTSTART;VALUE=DATE-TIME:20211101T160000Z
DTEND;VALUE=DATE-TIME:20211101T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
UID:fields-number-theory-seminar/52
DESCRIPTION:Title: Non-vanishing of Poincare series\nby Kumar 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 nu
mber of non-zero Poincare series for the full modular group and indicate h
ow it can be improved.\n
LOCATION:https://researchseminars.org/talk/fields-number-theory-seminar/52
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Stubley (McGill University)
DTSTART;VALUE=DATE-TIME:20211115T170000Z
DTEND;VALUE=DATE-TIME:20211115T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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://researchseminars.org/talk/fields-number-theory-seminar/53
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yves André (Institut de Mathématiques de Jussieu-Paris Rive Gauc
he and Sorbonne Université)
DTSTART;VALUE=DATE-TIME:20211206T170000Z
DTEND;VALUE=DATE-TIME:20211206T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T075057Z
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://researchseminars.org/talk/fields-number-theory-seminar/54
/
END:VEVENT
END:VCALENDAR