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BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART:20200409T203000Z
DTEND:20200409T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/1/">On the Kudla-Rapoport conjecture</a>\nby Chao Li (Columbia Universi
 ty) as part of Princeton/IAS number theory seminar\n\nLecture held in 214 
 Fine Hall (Princeton) or SH101 (IAS).\n\nAbstract\nThe Kudla-Rapoport conj
 ecture predicts a precise identity between the arithmetic intersection num
 ber of special cycles on unitary Rapoport-Zink spaces and the derivative o
 f local representation densities of hermitian forms. It is a key local ing
 redient to establish the arithmetic Siegel-Weil formula and the arithmetic
  Rallis inner product formula\, relating the height of special cycles on S
 himura varieties to the derivative of Siegel Eisenstein series and L-funct
 ions. We will motivate this conjecture\, explain a proof and discuss globa
 l applications.\n\nThis is joint work with Wei Zhang.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ana Caraiani (Imperial College\, London)
DTSTART:20200416T190000Z
DTEND:20200416T200000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/2/">Local-global compatibility in the crystalline case</a>\nby Ana Cara
 iani (Imperial College\, London) as part of Princeton/IAS number theory se
 minar\n\n\nAbstract\nLet F be a CM field. Scholze constructed Galois repre
 sentations associated to classes in the cohomology of locally symmetric sp
 aces for GL_n/F with p-torsion coefficients. These Galois representations 
 are expected to satisfy local-global compatibility at primes above p. Even
  the precise formulation of this property is subtle in general\, and uses 
 Kisin’s potentially semistable deformation rings. However\, this propert
 y is crucial for proving modularity lifting theorems. I will discuss joint
  work with J. Newton\, where we establish local-global compatibility in th
 e crystalline case under mild technical assumptions. This relies on a new 
 idea of using P-ordinary parts\, and improves on earlier results obtained 
 in joint work with P. Allen\, F. Calegari\, T. Gee\, D. Helm\, B. Le Hung\
 , J. Newton\, P. Scholze\, R. Taylor\, and J. Thorne in certain Fontaine-L
 affaille cases.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Thorne (University of Cambridge)
DTSTART:20200423T130000Z
DTEND:20200423T140000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/3/">Symmetric power functoriality for holomorphic modular forms</a>\nby
  Jack Thorne (University of Cambridge) as part of Princeton/IAS number the
 ory seminar\n\n\nAbstract\nLanglands’s functoriality conjectures predict
  the existence of “liftings” of automorphic representations along morp
 hisms of L-groups. A basic case of interest comes from the irreducible alg
 ebraic representations of GL(2)\, thought of as the L-group of the reducti
 ve group GL(2) over Q. I will discuss the proof\, joint with James Newton\
 ,  of the existence of the corresponding functorial liftings for a broad c
 lass of holomorphic modular forms\, including Ramanujan’s Delta function
 .\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henrik Gustafsson (IAS)
DTSTART:20200430T203000Z
DTEND:20200430T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/4/">Eulerianity of Fourier coefficients of automorphic forms</a>\nby He
 nrik Gustafsson (IAS) as part of Princeton/IAS number theory seminar\n\nLe
 cture held in 214 Fine Hall (Princeton) or SH101 (IAS).\n\nAbstract\nThe f
 actorization of Fourier coefficients of automorphic forms plays an importa
 nt role in a wide range of topics\, from the study of L-functions to the i
 nterpretation of scattering amplitudes in string theory.\n\nIn this talk I
  will present a transfer theorem which derives the Eulerianity of certain 
 Fourier coefficients from that of another coefficient. I will also discuss
  some applications of this theorem to Fourier coefficients of automorphic 
 forms in minimal and next-to-minimal representations.\n\nBased on recent w
 ork with Dmitry Gourevitch\, Axel Kleinschmidt\, Daniel Persson and Siddha
 rtha Sahi.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jayce Robert Getz (Duke University)
DTSTART:20200507T203000Z
DTEND:20200507T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/5/">On triple product L-functions</a>\nby Jayce Robert Getz (Duke Unive
 rsity) as part of Princeton/IAS number theory seminar\n\nLecture held in 2
 14 Fine Hall (Princeton) or SH101 (IAS).\n\nAbstract\nEstablishing the con
 jectured analytic properties of triple product L-functions is a crucial ca
 se of Langlands functoriality.  However\, little is known.  I will present
  work in progress on the case of triples of automorphic representations on
  GL_3\; in some sense this is the smallest case that appears out of reach 
 via standard techniques.  The approach is based on a the beautiful fibrati
 on method of Braverman and Kazhdan for constructing Schwartz spaces and pr
 oving analogues of the Poisson summation formula.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mladen Dimitrov (Université de Lille)
DTSTART:20200514T183000Z
DTEND:20200514T193000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/6/">A geometric view on Iwasawa theory</a>\nby Mladen Dimitrov (Univers
 ité de Lille) as part of Princeton/IAS number theory seminar\n\nLecture h
 eld in 214 Fine Hall (Princeton) or SH101 (IAS).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xin Wan (Morningside Center of Mathematics)
DTSTART:20200521T130000Z
DTEND:20200521T140000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/7/">Iwasawa theory and Bloch-Kato conjecture for unitary groups</a>\nby
  Xin Wan (Morningside Center of Mathematics) as part of Princeton/IAS numb
 er theory seminar\n\nLecture held in 214 Fine Hall (Princeton) or SH101 (I
 AS).\n\nAbstract\nWe describe a new method to study Eisenstein family and 
 Iwasawa theory on unitary groups over totally real fields of general signa
 tures. As a consequence we prove that if the central $L$-value of a cuspid
 al eigenform on the unitary group twisted by a CM character is 0\, then th
 e corresponding Selmer group has positive rank. The method also has a bypr
 oduct the $p$-adic functional equations for $p$-adic $L$-functions and $p$
 -adic families of Eisenstein series on unitary groups.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Farrell Brumley\, (Université Sorbonne Paris Nord)
DTSTART:20200528T140000Z
DTEND:20200528T150000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/8/">Joint equidistribution of adelic torus orbits and families of twist
 ed L-functions</a>\nby Farrell Brumley\, (Université Sorbonne Paris Nord)
  as part of Princeton/IAS number theory seminar\n\n\nAbstract\nThe classic
 al Linnik problems are concerned with the equidistribution of adelic torus
  orbits on the homogeneous spaces attached to inner forms of GL2\, as the 
 discriminant of the torus gets large. When specialized\, these problems ad
 mit beautiful classical interpretations\, such as the equidistribution of 
 integer points on spheres\, of Heegner points or packets of closed geodesi
 cs on the modular surface\, or of supersingular reductions of CM elliptic 
 curves. In the mid 20th century\, Linnik and his school established the eq
 uidistribution of many of these classical variants through his ergodic met
 hod\, under a congruence condition on the discriminants modulo a fixed aux
 iliary prime. More recently\, the Waldspurger formula and subconvex estima
 tes on L-functions were used to remove these congruence conditions\, and p
 rovide effective power-savings rates.\n\nIn their 2006 ICM address\, Miche
 l and Venkatesh proposed a variant of this problem in which one considers 
 the product of two distinct inner forms of GL2\, along with a diagonally e
 mbedded torus. One can again specialize the setting to obtain interesting 
 classical reformulations\, such as the joint equidistribution of integer p
 oints on the sphere\, together with the shape of the orthogonal lattice. T
 his hybrid context has received a great deal of attention recently in the 
 dynamics community\, where\, for instance\, the latter problem was solved 
 by Aka\, Einsiedler\, and Shapira\, under supplementary congruence conditi
 ons modulo two fixed primes\, using as critical input the joinings theorem
  of Einsiedler and Lindenstrauss.\n\nIn joint (ongoing) work with Valentin
  Blomer\, we remove the  supplementary congruence conditions in the joint 
 equidistribution problem\, conditionally on the Riemann Hypothesis\, while
  obtaining a logarithmic rate of convergence. The proof uses Waldsurger’
 s theorem and estimates of fractional moments of L-functions in the family
  of class group twists.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Richter (Northwestern University)
DTSTART:20200604T190000Z
DTEND:20200604T200000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/9/">Dynamical generalizations of the Prime Number Theorem and disjointn
 ess of additive and multiplicative actions</a>\nby Florian Richter (Northw
 estern University) as part of Princeton/IAS number theory seminar\n\n\nAbs
 tract\nOne of the fundamental challenges in number theory is to understand
  the intricate way in which the additive and multiplicative structures in 
 the integers intertwine. We will explore a dynamical approach to this topi
 c. After introducing a new dynamical framework for treating questions in m
 ultiplicative number theory\, we will present an ergodic theorem which con
 tains various classical number-theoretic results\, such as the Prime Numbe
 r Theorem\, as special cases. This naturally leads to a formulation of an 
 extended form of Sarnak's conjecture\, which deals with the disjointness o
 f actions of (N\,+) and (N\,*). This talk is based on joint work with Vita
 ly Bergelson.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rong Zhou (Imperial College London)
DTSTART:20200618T190000Z
DTEND:20200618T200000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/10/">Independence of $\\ell$ for Frobenius conjugacy classes attached t
 o abelian varieties</a>\nby Rong Zhou (Imperial College London) as part of
  Princeton/IAS number theory seminar\n\n\nAbstract\nLet $A$ be an abelian 
 variety over a number field $E\\subset \\mathbb{C}$ and let $v$ be a place
  of good reduction lying over a prime $p$. For a prime $\\ell\\neq p$\, a 
 result of Deligne implies that upon replacing $E$ by a finite extension\, 
 the Galois representation on the $\\ell$-adic Tate module of $A$ factors a
 s $\\rho_\\ell:\\mathrm{Gal}(\\overline{E}/E)\\rightarrow G_A$\, where $G_
 A$ is the Mumford--Tate group of $A_{\\mathbb{C}}$. For $p>2$\, we prove t
 hat the conjugacy class of  $\\rho_\\ell(\\mathrm{Frob}_v)$ is defined ove
 r $\\mathbb{Q}$ and independent of $\\ell$. This is joint work with Mark K
 isin.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vesselin Dimitrov (University of Toronto)
DTSTART:20200611T190000Z
DTEND:20200611T200000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/11/">New constraints on the Galois configurations of algebraic integers
  in the complex plane</a>\nby Vesselin Dimitrov (University of Toronto) as
  part of Princeton/IAS number theory seminar\n\n\nAbstract\nFekete (1923) 
 discovered the notion of transfinite diameter while studying the possible 
 configurations of Galois orbits of algebraic integers in the complex plane
 . Based purely on the fact that the discriminants of monic integer irreduc
 ible polynomials $P(X) \\in \\mathbb{Z}[X]$ are at least $1$ in magnitude 
 (since they are non-zero integers)\, he found that the incidences $(\\math
 cal{K}\, P)$ between these polynomials $P(X)$ and compacts $\\mathcal{K} \
 \subset \\mathbb{C}$ of transfinite diameter $d(\\mathcal{K}) < 1$ have fi
 nite fibers over the argument $\\mathcal{K}$. Here we say that $\\mathcal{
 K}$ and $P$ are in incidence if all the roots of $P$ belong to $\\mathcal{
 K}$. The descendants of Fekete's theorem are vast and powerful\, notably i
 ncluding the equidistribution theorems of Bilu\, Rumely and Szpiro-Ullmo-Z
 hang or -- in a different line of development -- the root separation bound
  of Mahler. But the input on the discriminant is sometimes too coarse to b
 e useful: in reality one expects\, but cannot prove\, that discriminants o
 f polynomials are large\, and dropping them by integrality is too crude in
  certain finer questions such as Lehmer's. \n\nBreusch (1951) solved the n
 on-reciprocal case of the Lehmer problem by taking up a lossless arithmeti
 c input from resultants rather than discriminants. In this talk\, I will p
 resent some further lossless constraints that derive from certain whole in
 finite sequences of Hankel determinants attached to the polynomial $P(X)$ 
 by algebraic operations. This will allow us to update on Fekete's theorem 
 on the incidences $(\\mathcal{K}\,P)$\, by focusing this time on the fiber
 s over the argument $P$ for compacts $\\mathcal{K}$ that are made of finit
 e unions of Jordan arcs continua covering the roots of $P$ with certain co
 ngruence conditions on $P$ and on the connected components of $\\mathcal{K
 }$. The ensuing taming on Galois orbits turn out to be sufficiently severe
  to resolve the conjecture of Schinzel and Zassenhaus (I will explain this
  case in detail)\, amidst certain other cases of the Lehmer problem that a
 re far off from Salem's extreme. In a geometric formulation for $\\mathcal
 {A}_g$ with its Kobayashi metric\, the root spacing constraints are likewi
 se sufficiently severe to furthermore yield the exact $\\mathcal{A}_g$ ana
 logs of the well-known polynomial counting theorems of Penner and Leininge
 r-Margalit on the "$L$-short" geodesics of moduli space $\\mathcal{M}_g$.\
 n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Peluse (IAS)
DTSTART:20200910T203000Z
DTEND:20200910T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/12/">An asymptotic version of the prime power conjecture for perfect di
 fference sets</a>\nby Sarah Peluse (IAS) as part of Princeton/IAS number t
 heory seminar\n\n\nAbstract\nA subset D of a finite cyclic group Z/mZ is c
 alled a "perfect difference set" if every nonzero element of Z/mZ can be w
 ritten uniquely as the difference of two elements of D. If such a set exis
 ts\, then a simple counting argument shows that m=n^2+n+1 for some nonnega
 tive integer n. Singer constructed examples of perfect difference sets in 
 Z/(n^2+n+1)Z whenever n is a prime power\, and it is an old conjecture tha
 t these are the only such n for which a perfect difference set exists. In 
 this talk\, I will discuss a proof of an asymptotic version of this conjec
 ture: the number of n less than N for which Z/(n^2+n+1)Z contains a perfec
 t difference set is ~N/log(N).\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT and IAS)
DTSTART:20200917T180000Z
DTEND:20200917T190000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/13/">Equivariant localization\, parity sheaves\, and cyclic base change
 </a>\nby Tony Feng (MIT and IAS) as part of Princeton/IAS number theory se
 minar\n\n\nAbstract\nLafforgue and Genestier-Lafforgue have constructed 
 the global and (semisimplified) local Langlands correspondences for arbit
 rary reductive groups over function fields. I will explain some recently 
 established properties of these correspondences regarding base change func
 toriality: existence of transfers for mod p automorphic forms through p-cy
 clic base change in the global correspondence\, and Tate cohomology realiz
 es p-cyclic base change in the mod p local correspondence. The proofs are 
 based on a combination of equivariant localization arguments (inspired by 
 work of Treumann-Venkatesh) and the theory of parity sheaves (due to Jutea
 u-Mautner-Williamson).\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhishek Oswal (IAS)
DTSTART:20200924T203000Z
DTEND:20200924T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/14/">A non-Archimedean definable Chow theorem</a>\nby Abhishek Oswal (I
 AS) as part of Princeton/IAS number theory seminar\n\n\nAbstract\nIn recen
 t years\, o-minimality has found some striking applications to diophantine
  geometry. The utility of o-minimal structures originates from the remarka
 bly tame topological properties satisfied by sets definable in such struct
 ures. Despite the rigidity that it imposes\, the theory is sufficiently fl
 exible to allow for a range of analytic constructions. An illustration of 
 this `tame' property is the following surprising generalization of Chow's 
 theorem proved by Peterzil and Starchenko -  A closed analytic subset of a
  complex algebraic variety that is also definable in an o-minimal structur
 e\, is in fact algebraic. While the o-minimal machinery aims to capture th
 e archimedean order topology of the real line\, it is natural to wonder if
  such a machinery can be set up over non-archimedean fields. In this talk\
 , we shall explore a non-archimedean analogue of an o-minimal structure an
 d a version of the definable Chow theorem in this context.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Fintzen (IAS and Duke University)
DTSTART:20201008T180000Z
DTEND:20201008T190000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/15/">Representations of p-adic groups and applications</a>\nby Jessica 
 Fintzen (IAS and Duke University) as part of Princeton/IAS number theory s
 eminar\n\n\nAbstract\nThe Langlands program is a far-reaching collection o
 f conjectures that relate different areas of mathematics including number 
 theory and representation theory. A fundamental problem on the representat
 ion theory side of the Langlands program is the construction of all (irred
 ucible\, smooth\, complex) representations of p-adic groups.\nI will provi
 de an overview of our understanding of the representations of p-adic group
 s\, with an emphasis on recent progress.\nI will also outline how new resu
 lts about the representation theory of p-adic groups can be used to obtain
  congruences between arbitrary automorphic forms and automorphic forms whi
 ch are supercuspidal at p\, which is joint work with Sug Woo Shin. This si
 mplifies earlier constructions of attaching Galois representations to auto
 morphic representations\, i.e. the global Langlands correspondence\, for g
 eneral linear groups. Moreover\, our results apply to general p-adic group
 s and have therefore the potential to become widely applicable beyond the 
 case of the general linear group.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carney Alexander (University of Rochester)
DTSTART:20201015T203000Z
DTEND:20201015T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/16
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/16/">Heights and dynamics over arbitrary fields</a>\nby Carney Alexande
 r (University of Rochester) as part of Princeton/IAS number theory seminar
 \n\n\nAbstract\n"Classically\, heights are defined over number fields or t
 ranscendence degree one function fields. This is so that the Northcott pro
 perty\, which says that sets of points with bounded height are finite\, ho
 lds. Here\, expanding on work of Moriwaki and Yuan-Zhang\, we show how to 
 define arithmetic intersections and heights relative to any finitely gener
 ated field extension K/k\, and construct canonical heights for polarizable
  arithmetic dynamical systems f:X->X. These heights have a corresponding N
 orthcott property when k is Q or F_q. When k is larger\, we show that Nort
 hcott for canonical heights is conditional on the non-isotriviality of f:X
 ->X\, generalizing work of Lang-Neron\, Baker\, and Chatzidakis-Hrushovski
 . Additionally\, we prove the Hodge Index Theorem for arithmetic intersect
 ions relative to K/k. Since\, when Northcott holds\, preperiodic points ar
 e the same as height zero points\, this has applications to dynamical syst
 ems. By the Lefschetz principle\, these results can be applied over any fi
 eld.\n"\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tasho Kaletha (University of Michigan and IAS)
DTSTART:20201029T203000Z
DTEND:20201029T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/17
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/17/">An explicit supercuspidal local Langlands correspondence</a>\nby T
 asho Kaletha (University of Michigan and IAS) as part of Princeton/IAS num
 ber theory seminar\n\n\nAbstract\nWe will give an explicit construction an
 d description of a supercuspidal local Langlands correspondence for any p-
 adic group G that splits over a tame extension\, provided p does not divid
 e the order of the Weyl group. This construction matches any discrete Lang
 lands parameters with trivial monodromy to an L-packet consisting of super
 cuspidal representations\, and describes the internal structure of these L
 -packets.\n\nThe construction has two parts. The depth-zero part involves 
 generalizing to disconnected groups results of Lusztig on the decompositio
 n of a non-singular Deligne-Lusztig induction. Higher multiplicities occur
  in this decomposition and are handled using work of Bonnafe-Dat-Rouquier.
  The positive-depth part involves functorial transfer from a twisted Levi 
 subgroup\, which is made possible by an improvement of Yu's construction o
 f supercuspidal representations obtained in recent joint work with Fintzen
  and Spice\, and consideration of Harish-Chandra characters.\n\nWe will al
 so discuss ongoing work towards related conjectures: Shahidi's generic L-p
 acket conjecture\, Hiraga-Ichino-Ikeda formal degree conjecture\,  stabili
 ty and endoscopic transfer.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shai Evra (Princeton)
DTSTART:20201119T213000Z
DTEND:20201119T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/18
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/18/">Ramanujan Conjecture and the Density Hypothesis</a>\nby Shai Evra 
 (Princeton) as part of Princeton/IAS number theory seminar\n\n\nAbstract\n
 The Generalized Ramanujan Conjecture (GRC) for GL(n) is a central open pro
 blem in modern number theory. Its resolution is known to yield several imp
 ortant applications. For instance\, the Ramanujan-Petersson conjecture for
  GL(2)\, proven by Deligne\, was a key ingredient in the work of Lubotzky-
 Phillips-Sarnak on Ramanujan graphs.\nOne can also state analogues (Naive)
  Ramanujan Conjectures (NRC) for other reductive groups. However\, in the 
 70's Kurokawa and Howe-Piatetski-Shapiro proved that the (NRC) fails even 
 for quasi-split classical groups.\nIn the 90's Sarnak-Xue put forth a Dens
 ity Hypothesis version of the (NRC)\, which serves as a replacement of the
  (NRC) in applications.\nIn this talk I will describe a possible approach 
 to proving the Density Hypothesis for definite classical groups\, by invok
 ing deep and recent results coming from the Langlands program: The endosco
 pic classification of automorphic representations of classical groups due 
 to Arthur\, and the proof of the Generalized Ramanujan-Petersson Conjectur
 e.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Chen (Columbia University)
DTSTART:20201105T213000Z
DTEND:20201105T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/19
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/19/">Strong approximation for the Markoff equation via nonabelian level
  structures on elliptic curves</a>\nby William Chen (Columbia University) 
 as part of Princeton/IAS number theory seminar\n\n\nAbstract\nFollowing Bo
 urgain\, Gamburd\, and Sarnak\, we say that the Markoff equation x^2 + y^2
  + z^2 - 3xyz = 0 satisfies strong approximation at a prime p if its integ
 ral points surject onto its F_p points. In 2016\, Bourgain\, Gamburd\, and
  Sarnak were able to establish strong approximation at all but a sparse (b
 ut infinite) set of primes\, and conjecture that it holds at all primes. B
 uilding on their results\, in this talk I will explain how to obtain stron
 g approximation for all but a finite and effectively computable set of pri
 mes\, thus reducing the conjecture to a finite computation. The key result
  amounts to establishing a congruence on the degree of a certain line bund
 le on the moduli stack of elliptic curves with SL(2\,p)-structures. To mak
 e contact with the Markoff equation\, we use the fact that the Markoff sur
 face is a level set of the character variety for SL(2) representations of 
 the fundamental group of a punctured torus\, and that the strong approxima
 tion conjecture can be expressed in terms of the mapping class group actio
 n on the character variety\, which in turn also determines the geometry of
  the moduli stack of elliptic curves with SL(2\,p)-structures. As time all
 ows we will also describe a number of applications.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpoge (Columbia University)
DTSTART:20201112T213000Z
DTEND:20201112T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/20
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/20/">Effective height bounds for odd-degree totally real points on some
  curves</a>\nby Levent Alpoge (Columbia University) as part of Princeton/I
 AS number theory seminar\n\n\nAbstract\nLet $\\mathfrak o$ be an order in 
 a totally real field\, say $F$. Let $K$ be an odd-degree totally real fiel
 d. Let $S$ be a finite set of places of $K$. We study $S$-integral $K$-poi
 nts on integral models $H_\\mathfrak o$ of Hilbert modular varieties becau
 se not only do said varieties admit complete curves (thus reducing questio
 ns about such curves' $K$-rational points to questions about $S$-integral 
 $K$-points on these integral models)\, they also have their $S$-integral $
 K$-points controlled by known cases of modularity\, in the following way. 
 First assume for clarity modularity of all $\\GL_2$-type abelian varieties
  over $K$ --- then all $S$-integral $K$-points on $H_\\mathfrak o$ arise f
 rom $K$-isogeny factors of the $[F:\\mathbb Q]$-th power of the Jacobian o
 f a single Shimura curve with level structure (by Jacquet-Langlands transf
 er). By a generalization of an argument of von Känel\, isogeny estimates 
 of Raynaud/Masser-Wüstholz and Bost's lower bound on the Faltings height 
 suffice to then bound the heights of all points in $H_\\mathfrak o(\\mathf
 rak o_{K\,S})$. As for the assumption\, though modularity is of course not
  known in this generality\, by following Taylor's (sufficiently explicit f
 or us) proof of his potential modularity theorem we are able to make the a
 bove unconditional.\n\nFinally we use the hypergeometric abelian varieties
  associated to the arithmetic triangle group $\\Delta(3\,6\,6)$ to give ex
 plicit examples of curves to which the above height bounds apply. Specific
 ally\, we prove that\, for $a\\in \\overline{\\mathbb Q}^\\times$ totally 
 real of odd degree (e.g. $a = 1$) and $L/\\mathbb Q(a)$ totally real of od
 d degree\, there is an effectively computable $c = c_{a\,L}\\in \\Z^+$ suc
 h that all $x\,y\\in L$ satisfying $x^6 + 4y^3 = a^2$ satisfy $h(x) < c$. 
 Note that this gives infinitely many curves for each of which Faltings' th
 eorem is now effective over infinitely many number fields.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jingwei Xiao (Princeton University and IAS)
DTSTART:20201203T213000Z
DTEND:20201203T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/21
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/21/">A unitary analogy of Friedberg-Jacquet and Guo-Jacquet periods and
  central values of standard L functions on GL(2n)</a>\nby Jingwei Xiao (Pr
 inceton University and IAS) as part of Princeton/IAS number theory seminar
 \n\n\nAbstract\nLet G be a reductive group over a number field F and H a s
 ubgroup. Automorphic periods study the integrals of cuspidal automorphic f
 orms on G over H(F)\\H(A_F). They are often related to special values of c
 ertain L functions. One of the most notable case is when (G\,H)=(U(n+1)☓
 U(n)\, U(n))\, and these periods are related to central values of Rankin-S
 elberg L functions on GL(n+1)☓GL(n). In this talk\, I will explain my wo
 rk in progress with Wei Zhang that studies central values of standard L fu
 nctions on GL(2n) using (G\,H)=(U(2n)\, U(n)☓U(n)) and some variants. I 
 shall explain the conjecture and a relative trace formula approach to stud
 y it. We prove the required fundamental lemma using a limit of the Jacquet
 -Rallis fundamental lemma and Hironaka’s characterization of spherical f
 unctions on the space of non degenerate Hermitian matrices. Also\, the que
 stion admits an arithmetic analogy.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joni Teravainen (Oxford)
DTSTART:20201210T213000Z
DTEND:20201210T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/22
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/22/">On the Liouville function at polynomial arguments</a>\nby Joni Ter
 avainen (Oxford) as part of Princeton/IAS number theory seminar\n\n\nAbstr
 act\nLet λ be the Liouville function and P(x) any polynomial that is not 
 a square. An open problem formulated by Chowla and others asks to show tha
 t the sequence λ(P(n)) changes sign infinitely often. We present a soluti
 on to this problem for new classes of polynomials P\, including any produc
 t of linear factors or any product of quadratic factors of a certain type.
  The proofs also establish some nontrivial cancellation in Chowla and Elli
 ott type correlation averages.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lue Pan (University of Chicago)
DTSTART:20201022T203000Z
DTEND:20201022T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/23
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/23/">On the locally analytic vectors of the completed cohomology of mod
 ular curves</a>\nby Lue Pan (University of Chicago) as part of Princeton/I
 AS number theory seminar\n\n\nAbstract\nA classical result identifies holo
 morphic modular forms with highest weight vectors of certain representatio
 ns of SL_2(\\mathbb{R}). We study locally analytic vectors of the (p-adica
 lly) completed cohomology of modular curves and prove a p-adic analogue of
  this result. As applications\, we are able to prove a classicality result
  for overconvergent eigenforms and give a new proof of Fontaine-Mazur conj
 ecture in the irregular case under some mild hypothesis. One technical too
 l is relative Sen theory which allows us to relate infinitesimal group act
 ion with Hodge(-Tate) structure.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joel Nagloo (City University of New York)
DTSTART:20210121T213000Z
DTEND:20210121T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/24
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/24/">Ax-Lindemann-Weierstrass Theorem (ALW) for Fuchsian automorphic fu
 nctions</a>\nby Joel Nagloo (City University of New York) as part of Princ
 eton/IAS number theory seminar\n\n\nAbstract\nOver the last decades\, foll
 owing works around the Pila-Wilkie counting theorem in the context of o-mi
 nimality\, there has been a surge in interest around functional transcende
 nce results\, in part due to their connection with special points conjectu
 res. A prime example is Pila's modular ALW Theorem and its role in his pro
 of of the André-Oort conjecture. \n\nIn this talk we will discuss how an 
 entirely new approach\, using the model theory of differential fields\, ca
 n be used to prove the Ax-Lindemann-Weierstrass (ALW) Theorem with derivat
 ives for Fuchsian automorphic functions - a direct generalization of Pila
 ’s ALW theorem. We will also explain how new cases of the André-Pink co
 njecture can be obtained using this new approach. This is joint work with 
 G. Casale and J. Freitag.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathilde Gerbelli-Gauthier (IAS)
DTSTART:20210211T213000Z
DTEND:20210211T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/25
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/25/">Cohomology of Arithmetic Groups and Endoscopy</a>\nby Mathilde Ger
 belli-Gauthier (IAS) as part of Princeton/IAS number theory seminar\n\n\nA
 bstract\nHow fast do Betti numbers grow in a congruence tower of compact a
 rithmetic manifolds? The dimension of the middle degree of cohomology is p
 roportional to the volume of the manifold\, but away from the middle the g
 rowth is known to be sub-linear in the volume. I will explain how automorp
 hic representations and the phenomenon of endoscopy provide a framework to
  understand and quantify this slow growth. Specifically\, I will discuss h
 ow to obtain some explicit bounds in the case of unitary groups using Arth
 ur’s stable trace formula.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (IAS)
DTSTART:20210218T213000Z
DTEND:20210218T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/26
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/26/">Exceptional jumps of Picard rank of K3 surfaces over number fields
 </a>\nby Salim Tayou (IAS) as part of Princeton/IAS number theory seminar\
 n\n\nAbstract\nGiven a K3 surface X over a number field K\, we prove that 
 the set of primes of K where the geometric Picard rank jumps is infinite\,
  assuming that X has everywhere potentially good reduction. This result is
  formulated in the general framework of GSpin Shimura varieties and I will
  explain other applications to abelian surfaces. I will also discuss appli
 cations to the existence of rational curves on K3 surfaces. The results in
  this talk are joint work with Ananth Shankar\, Arul Shankar and Yunqing T
 ang.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hang Xue (The University of Arizona)
DTSTART:20210325T203000Z
DTEND:20210325T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/27
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/27/">The local Gan-Gross-Prasad conjecture for real unitary groups</a>\
 nby Hang Xue (The University of Arizona) as part of Princeton/IAS number t
 heory seminar\n\n\nAbstract\nA classical branching theorem of Weyl describ
 es how an irreducible representation of compact U(n+1) decomposes when res
 tricted to U(n). The local Gan-Gross-Prasad conjecture provides a conjectu
 ral extension to the setting of representations of noncompact unitary grou
 ps lying in a generic L-packet. We prove this conjecture. Previously Beuza
 rt-Plessis proved the ''multiplicity one in a Vogan packet'' part of the c
 onjecture for tempered L-packets using the local trace formula approach in
 itiated by Waldspurger. Our proof uses theta lifts instead\, and is indepe
 ndent of the trace formula argument.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Mundy (Columbia University)
DTSTART:20210401T203000Z
DTEND:20210401T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/28
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/28/">Eisenstein series\, $p$-adic deformations\, Galois representations
 \, and the group $G_2$.</a>\nby Sam Mundy (Columbia University) as part of
  Princeton/IAS number theory seminar\n\n\nAbstract\nI will explain some re
 cent work on special cases of the Bloch-Kato conjecture for the symmetric 
 cube of certain modular Galois representations. Under certain standard con
 jectures\, this work constructs nontrivial elements in the Selmer groups o
 f these symmetric cube Galois representations\; this works by $p$-adically
  deforming critical Eisenstein series in a generically cuspidal family of 
 automorphic representations\, and then constructing a lattice in the assoc
 iated family of Galois representations\, all for the exceptional group $G_
 2$. While I will touch on all of these aspects of the construction\, I wil
 l mainly focus on the Galois side in this talk.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART:20210415T203000Z
DTEND:20210415T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/29
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/29/">Beilinson-Bloch conjecture for unitary Shimura varieties</a>\nby C
 hao Li (Columbia University) as part of Princeton/IAS number theory semina
 r\n\n\nAbstract\nFor certain automorphic representations $\\pi$ on unitary
  groups\, we show that if $L(s\, \\pi)$ vanishes to order one at the cente
 r $s=1/2$\, then the associated $\\pi$-localized Chow group of a unitary S
 himura variety is nontrivial. This proves part of the Beilinson-Bloch conj
 ecture for unitary Shimura varieties\, which generalizes the BSD conjectur
 e. Assuming Kudla's modularity conjecture\, we further prove the arithmeti
 c inner product formula for $L'(1/2\, \\pi)$\, which generalizes the Gross
 -Zagier formula. We will motivate these conjectures and discuss some aspec
 ts of the proof. We will also mention recent extensions applicable to symm
 etric power L-functions of elliptic curves. This is joint work with Yifeng
  Liu.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yves André (CNRS)
DTSTART:20210429T200000Z
DTEND:20210429T210000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/30
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/30/">On the canonical\, fpqc and finite topologies: classical questions
 \, new answers (and conversely)</a>\nby Yves André (CNRS) as part of Prin
 ceton/IAS number theory seminar\n\n\nAbstract\nUp to a finite covering\, a
  sequence of nested subvarieties of an affine algebraic variety just looks
  like a flag of vector spaces (Noether)\; understanding this « up to » i
 s a primary motivation for a fine study of finite coverings. \n\nThe aim o
 f this talk is to give a bird-eye view of some fundamental questions about
  them\, which took root in Algebraic Geometry (descent problems etc.)\, th
 en motivated major trends in Commutative Algebra (F-singularities etc.)\, 
 and recently found complete solutions using p-adic methods (perfectoids). 
 Rather than going into detail of the latter\, the emphasis will be on synt
 hesizing\, from the geometric viewpoint\, a rather scattered theme. \n\nTh
 is is based on joint work with Luisa Fiorot (https://doi.org/10.2422/2036-
 2145.201912_006)\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gal Dor (Tel Aviv University)
DTSTART:20210304T213000Z
DTEND:20210304T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/31
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/31/">Monoidal Structures on GL(2)-Modules and Abstractly Automorphic Re
 presentations</a>\nby Gal Dor (Tel Aviv University) as part of Princeton/I
 AS number theory seminar\n\n\nAbstract\nConsider the function field F of a
  smooth curve over $F_q$\, with q>2.\n\nL-functions of automorphic represe
 ntations of GL(2) over F are important objects for studying the arithmetic
  properties of the field F. Unfortunately\, they can be defined in two dif
 ferent ways: one by Godement-Jacquet\, and one by Jacquet-Langlands. Class
 ically\, one shows that the resulting L-functions coincide using a complic
 ated computation.\n\nEach of these L-functions is the GCD of a family of z
 eta integrals associated to test data. I will categorify the question\, by
  showing that there is a correspondence between the two families of zeta i
 ntegrals\, instead of just their L-functions. The resulting comparison of 
 test data will induce an exotic symmetric monoidal structure on the catego
 ry of representations of GL(2).\n\nIt turns out that an appropriate space 
 of automorphic functions is a commutative algebra with respect to this sym
 metric monoidal structure. I will outline this construction\, and show how
  it can be used to construct a category of automorphic representations.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandru Zaharescu (UIUC)
DTSTART:20210311T213000Z
DTEND:20210311T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/32
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/32/">Some remarks on Landau-Siegel zeros</a>\nby Alexandru Zaharescu (U
 IUC) as part of Princeton/IAS number theory seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia university)
DTSTART:20210318T203000Z
DTEND:20210318T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/33
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/33/">The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties<
 /a>\nby Will Sawin (Columbia university) as part of Princeton/IAS number t
 heory seminar\n\n\nAbstract\nFaltings proved the statement\, previously co
 njectured by Shafarevich\, that there are finitely many abelian varieties 
 of dimension n\, defined over a fixed number field\, with good reduction o
 utside a fixed finite set of primes\, up to isomorphism. In joint work wit
 h Brian Lawrence\, we prove an analogous finiteness statement for hypersur
 faces in a fixed abelian variety with good reduction outside a finite set 
 of primes. I will give a broad introduction to some of the ideas in the pr
 oof\, which builds on p-adic Hodge theory techniques from work of Lawrence
  and Venkatesh as well as sheaf convolution in algebraic geometry.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Harper (University of Warwick)
DTSTART:20210408T203000Z
DTEND:20210408T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/34
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/34/">Low moments of character sums</a>\nby Adam Harper (University of W
 arwick) as part of Princeton/IAS number theory seminar\n\n\nAbstract\nSums
  of Dirichlet characters $\\sum_{n \\leq x}\\chi(n)$ (where $\\chi$ is a c
 haracter modulo some prime $r$\, say) are one of the best studied objects 
 in analytic number theory. Their size is the subject of numerous results a
 nd conjectures\, such as the P\\'olya--Vinogradov inequality and the Burge
 ss bound. One way to get information about this is to study the power mome
 nts $\\frac{1}{r−1}\\sum_{\\chi\\mod r}|\\sum_{n≤x}\\chi(n)|^{2q}$\, w
 hich turns out to be quite a subtle question that connects with issues in 
 probability and physics. In this talk I will describe an upper bound for t
 hese moments when $0≤q≤1$. I will focus mainly on the number theoretic
  issues arising.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Rapinchuk (University of Virginia)
DTSTART:20210506T203000Z
DTEND:20210506T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/35
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/35/">Groups with bounded generation: old and new</a>\nby Andrei Rapinch
 uk (University of Virginia) as part of Princeton/IAS number theory seminar
 \n\n\nAbstract\nA group is said to have bounded generation (BG) if it is a
  finite product of cyclic subgroups. We will survey the known examples of 
 groups with (BG) and their properties. We will then report on a recent res
 ult (joint with P. Corvaja\, J. Ren and U. Zannier) that non-virtually abe
 lian anisotropic linear groups (i. e. those consisting entirely of semi-si
 mple elements) are not boundedly generated. The proofs rely on number-theo
 retic techniques.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART:20210422T203000Z
DTEND:20210422T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/36
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/36/">Kolyvagin's conjecture and higher congruences of modular forms</a>
 \nby Naomi Sweeting (Harvard) as part of Princeton/IAS number theory semin
 ar\n\n\nAbstract\nGiven an elliptic curve E\, Kolyvagin used CM points on 
 modular curves to construct a system of classes valued in the Galois cohom
 ology of the torsion points of E. Under the conjecture that not all of the
 se classes vanish\, he gave a description for the Selmer group of E. This 
 talk will report on recent work proving new cases of Kolyvagin's conjectur
 e. The proof builds on work of Wei Zhang\, who used congruences between mo
 dular forms to prove Kolyvagin's conjecture under some technical hypothese
 s. We remove many of these hypotheses by considering congruences modulo hi
 gher powers of p. The talk will explain the difficulties associated with h
 igher congruences of modular forms and how they can be overcome.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Smith (MIT)
DTSTART:20210225T213000Z
DTEND:20210225T223000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/37
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/37/">Selmer groups and a Cassels-Tate pairing for finite Galois modules
 </a>\nby Alexander Smith (MIT) as part of Princeton/IAS number theory semi
 nar\n\n\nAbstract\nI will discuss some new results on the structure of Sel
 mer groups of finite Galois modules over global fields. Tate's definition 
 of the Cassels-Tate pairing can be extended to a pairing on such Selmer gr
 oups with little adjustment\, and many of the fundamental properties of th
 e Cassels-Tate pairing can be reproved with new methods in this setting. I
  will also give a general definition of the theta/Mumford group and relate
  it to the structure of the Cassels-Tate pairing\, generalizing work of Po
 onen and Stoll.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maksym Radziwiłł (Caltech)
DTSTART:20210513T203000Z
DTEND:20210513T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/38
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/38/">Expansion and parity</a>\nby Maksym Radziwiłł (Caltech) as part 
 of Princeton/IAS number theory seminar\n\n\nAbstract\nI will discuss recen
 t work with Harald Helfgott in which we establish roughly speaking that th
 e graph connecting n to n +/- p with p a prime dividing n is almost "local
 ly Ramanujan". As a result we obtain improvements of results of Tao and Ta
 o-Teravainen on logarithmic Chowla. I will discuss the main ideas in the p
 roof and the connections with logarithmic Chowla.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Larsen (Indiana University Bloomington)
DTSTART:20210527T203000Z
DTEND:20210527T213000Z
DTSTAMP:20260422T225845Z
UID:PrincetonIASNT/39
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PrincetonIAS
 NT/39/">Character estimates for classical finite simple groups</a>\nby Mic
 hael Larsen (Indiana University Bloomington) as part of Princeton/IAS numb
 er theory seminar\n\n\nAbstract\nThis is intended to complement the recent
  talk of Pham Huu Tiep in this seminar but will not assume familiarity wit
 h that talk.  The estimates in the title are upper bounds of the form $|\\
 chi(g)| \\le \\chi(1)^\\alpha$\, where $\\chi$ is irreducible and $\\alpha
 $ depends on the size of the centralizer of g.  I will briefly discuss geo
 metric applications of such bounds\, explain how probability theory can be
  used to reduce to the case of elements g of small centralizer\, discuss t
 he level theory of characters\, and conclude with the reduction to the cas
 e of characters $\\chi$ of large degree.  For such pairs (g\,$\\chi$)\, ex
 ponential character bounds are trivial.\n
LOCATION:https://researchseminars.org/talk/PrincetonIASNT/39/
END:VEVENT
END:VCALENDAR