Academic year 2019-20

September 17, 2019

Zheng Liu (UC Santa Barbara): "Doubling archimedean zeta integrals for symplectic and unitary groups"

Abstract: In order to verify the compatibility between the conjecture of Coates--Perrin-Riou and the interpolation results of the p-adic L-functions constructed by using the doubling method, a doubling archimedean zeta integral needs to be calculated for holomorphic discrete series. When the holomorphic discrete series is of scalar weight, it has been done by Bocherer--Schmidt and Shimura. I will explain a way to compute this archimedean zeta integral for general vector weights by using the theory of theta correspondence.

September 17, 2019

Will Sawin (Columbia/ Clay Mathematics Institute): "The sup-norm problem for modular forms over function fields and geometry"

Abstract: The sup-norm problem is a concrete question about automorphic forms, which asks for bounds on their largest value (when viewed as a function on a modular curve, Shimura variety, or locally symmetry space.) We describe an approach to the analogous problem in the function field setting using geometric techniques, which we carry out in a special case. This involves, following Drinfeld, viewing the automorphic form as arising from a geometric object (a perverse sheaf) and then studying this object using techniques adapted from the theory of perverse sheaves on complex varieties, which reduce the problem to an elementary calculation.

October 29, 2019

Aaron Pollack (Duke): "Modular forms on exceptional groups"

Abstract: By a "modular form" for a reductive group G we mean an automorphic form that has some sort of very nice Fourier expansion. The classic example are the holomorphic Siegel modular forms, which are special automorphic functions for the group Sp_{2g}. Following work of Gan, Gross, Savin, and Wallach, it turns out that there is a notion of modular forms on certain real forms of the exceptional groups. I will define these objects and explain what is known about them.

October 29, 2019

Kirsten Wickelgren (Duke): "An arithmetic count of rational plane curves"

Abstract: Over the complex numbers, the number of degree d rational plane curves passing through 3d-1 points is independent of (generically) chosen points, and is computed recursively with a celebrated formula due to Kontsevich. Over other fields, this number is finite but can vary. For example, there can be 8, 10, or 12 real degree 3 curves through 8 real points. Over the real numbers, Welschinger found an invariant count by weighting real rational curves by +1/-1 when the number of non-split real nodes is even/ odd. But what about over fields like Q, Qp, Fp? It is a feature of A1-homotopy theory that analogous real and complex results can indicate the presence of a common generalization, valid over a general field. We give such a generalization, weighting by the discriminant of the double points, giving a count valued in bilinear forms. No familiarity of A1-homotopy theory will be assumed. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon.

December 3, 2019

Bao Le Hung (Northwestern): "Serre weights, Breuil-Mezard cycles and deformed affine Springer fibers"

Abstract: Given a locally symmetric space and a mod p system of Hecke eigenvalues, the weight part of Serre's conjecture seeks to classify the cohomology groups of the locally symmetric space where this system of eigenvalues contribute. Via Taylor-Wiles patching, this problem is known to be closely linked to the Breuil-Mezard conjecture, which predicts a representation theoretic recipe to build mod p fibers of local Galois deformation rings (and more generally, the Emerton-Gee moduli stack of local Galois representation) out of atomic pieces corresponding to Serre weights. I will describe progress on both questions in many cases, via a theory of local models, which relates (portions of) the Emerton-Gee stack to mixed characteristics versions of (deformed) affine Springer fibers.

December 3, 2019

Shou-Wu Zhang (Princeton): "On modularity of Kudla’s generating series of 0-cycles”

Abstract: On some Shimura varieties of orthogonal or unitary type, Steve Kudla has defined some generating series of special cycles and conjectured their modularity as Siegel modular forms. In this talk, I will describe a proof of this modularity for zero cycles modulo Abel-Jacobi kernel.

February 4, 2020

Wei Ho (Michigan): The Hasse principle for some genus one curves

Abstract: We will discuss problems related to the Hasse principle for some classes of varieties, with a special focus on genus one curves given by bihomogeneous polynomials of bidegree (2,2) in $\mathbb{P}^1 \times \mathbb{P}^1$. For example, we will describe how to compute the proportion of these curves that are everywhere locally soluble (joint work with Tom Fisher and Jennifer Park), and we show that the Hasse principle fails for a positive proportion of these curves, by comparing the average sizes of 2- and 3-Selmer groups for a family of elliptic curves with a marked point (joint work with Manjul Bhargava).

February 4, 2020

Marty Weissman (UC Santa Cruz): The arithmetic of arithmetic hyperbolic Coxeter groups

Abstract: In the 1990s, John H. Conway developed a visual approach to the study of integer-valued binary quadratic forms. His creation, the "topograph," sheds light on classical reduction theory, the solution of Pell-type equations, and allows tedious algebraic estimates to be simplified with straightforward geometric arguments. The geometry of the topograph arises from a Coxeter group of type (3, infinity) and its close relation to the group PGL(2,Z). From this perspective, the insights of Conway arise from an arithmetic hyperbolic Coxeter group. In this talk, I will survey Conway's approach to integer binary quadratic forms, and show how similar techniques can yield number theoretic results from other arithmetic hyperbolic Coxeter groups. These results are joint work with Chris D. Shelley and Suzana Milea.

February 25, 2020

Daniel Litt (U. Georgia): The section conjecture at the boundary of moduli space

Abstract: Grothendieck's section conjecture predicts that over arithmetically interesting fields (e.g. number fields or p-adic fields), rational points on a smooth projective curve X of genus at two can be detected via the arithmetic of the etale fundamental group of X. We construct infinitely many curves of each genus satisfying the section conjecture in interesting ways, building on work of Stix, Harari, and Szamuely. The main input to our result is an analysis of the degeneration of certain torsion cohomology classes on the moduli space of curves at various boundary components. This is (preliminary) joint work with Padmavathi Srinivasan, Wanlin Li, and Nick Salter.

February 25, 2020

Chao Li (Columbia): On the Kudla-Rapoport conjecture

Abstract: The Kudla-Rapoport conjecture predicts a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport-Zink spaces and the derivatives of local representation densities of hermitian forms. It is a key local ingredient to establish the arithmetic Siegel-Weil formula, relating the height of generating series of special cycles on Shimura varieties to the derivative of Eisenstein series. We discuss a proof of this conjecture and global applications. This is joint work with Wei Zhang.

April 7, 2020

Hang Xue (Arizona): Talk cancelled

Abstract: TBA

April 7, 2020

Jacob Tsimerman (Toronto): Talk cancelled

Abstract: TBA

Academic year 2018-19

October 16, 2018

Rachel Ollivier (UBC): A derived Hecke algebra in the context of the mod p Langlands program

Abstract: Given a p-adic reductive group G and its (pro-p) Iwahori-Hecke algebra H, we are interested in the link between the category of smooth representations of G and the category of H-modules. When the field of coefficients has characteristic zero this link is well understood by work of Bernstein and Borel. In characteristic p things are still poorly understood. In this case the role of the pro-p Iwahori-Hecke algebra H is played by a differential graded Hecke algebra. In particular, by work of Peter Schneider, the module category over the d.g. Hecke algebra is equivalent to the derived category of smooth representations of G. Unlike in the case of H, we know little about the structure of this d.g. Hecke algebra. In this talk I will report on joint work with Peter Schneider where we take the first steps in this direction by studying the cohomology of the d.g. Hecke algebra.

October 16, 2018

Xinwen Zhu (Caltech): Selmer groups for certain Rankin-Selberg L-functions

Abstract: To bound the Bloch-Kato Selmer group of certain Rankin-Selberg L-functions, we construct a Bertonili-Darmon-Kolyvagin system using the Gan-Gross-Prasad cycles on the product of unitary Shimura varieties. This allows us to relate the trivality/non-trivialty of the Selmer group to the vanishing order of the L-function. We discuss some examples from elliptic curves. This is a joint work with Yifeng Liu, Yichao Tiao, Liang Xiao and Wei Zhang.

November 6, 2018

Manish Patnaik (University of Alberta): Adjoint L-functions and Loop Groups over Function Fields

Abstract: In the function field setting, we will explain how the adjoint L-functions for cusp forms on finite-dimensional groups appear in a functional equation relating two rather different Eisenstein series on an infinite-dimensional loop group. One of these Eisentein series turns out to be holomorphic and the other, introduced by A. Braverman and D. Kazhdan, needs a "regularization" in order to be defined and is conjectured to extend to a meromorphic function on some half space. This is joint work with H. Garland and S.D. Miller.

November 6, 2018

Akshay Venkatesh (Stanford/IAS): Heights of motives and heights of modular forms

Abstract: Kato has defined a notion of height for any motive, generalizing the Faltings height of an abelian variety. I will review this definition. Now, for those motives that are related to the cohomology of arithmetic groups, I will define another "automorphic" height. The definitions are very different, but surprisingly the two notions of height appear to be closely related.

December 11, 2018

Jayce Getz (Duke University): On triple product L functions for GL(3)

Abstract: I will define a multiple Dirichlet series (MDS) attached to three automorphic representations of GL(3) and a quadruple of Hecke characters. Joint work with B. Liu on summation formulae for triples of quadratic spaces implies that the MDS admits meromorphic continuations in all variables and satisfies several functional equations. I will then explain work in progress relating one Dirichlet series in the MDS to the triple product L function attached to the three automorphic representations and how it can be used to deduce new information about this L-function.

December 11, 2018

Brian Lawrence (University of Chicago): Bounding points on curves using p-adic hodge theory

Abstract: Methods of p-adic analysis provide the most powerful tools to bound the set of rational points on a curve. The earliest work in this direction was the method of Chabauty; in many cases this is already enough to enumerate (with proof) the rational points. Much more recently, work of Kim on the unipotent fundamental group has led to computational breakthroughs by Balakrishnan, Dogra et al. The method of Chabauty works only when r < g (here $r$ is the rank of the Jacobian, and $g$ the genus of the curve); Kim's method has been applied to the case r = g, though it is expected to apply in general. I will discuss a new method for bounding points on curves, using instead a reductive representation of the fundamental group. This new method may apply to all curves, but it presents substantial computational difficulties.

February 12, 2019

A.C. Cojocaru (University of Illinois at Chicago): Questions about the reductions modulo primes of an elliptic curve

Abstract: Many remarkable questions about prime numbers have natural analogues in the context of elliptic curves. Among them, Artin's primitive root conjecture, the twin prime conjecture, and the Schinzel hypothesis have inspired a broad family of conjectures regarding the behavior of the reductions modulo primes of an elliptic curve over Q. I will give an overview of such questions and progress made towards their resolution.

February 12, 2019

Dino Lorenzini (University of Georgia): New points on curves

Abstract: Let K be any field. Given an algebraic variety X/K and a field extension L/K, a point in X(L) is called a new point of X/K over L if the coordinates of the point generate the field L/K. In other words, the point of X(L) is a new point if it corresponds to a closed point of X whose residue field is isomorphic to L. Many classical problems in arithmetic geometry fix a variety X/K and ask for information on the sets X(L) as L/K varies. In this talk we will turn the question around, fix an extension L/K, and consider the smooth projective curves X/K with a new point over L.

April 9, 2019

Shuichiro Takeda (University of Missouri): Subrepresentation theorems for p-adic symmetric spaces

Abstract: In this talk, I will talk about a couple of subrepresentation theorems for p-adic symmetric spaces, which can be considered as a generalization of well-known subrepresentations for admissible representations. The idea is to generalize the theory developed by Kato and Takano.

April 9, 2019

Kumar Murty (University of Toronto): Kummer's conjecture

Abstract: Kummer's conjecture on the growth of the minus part of the class number of a prime cyclotomic field is widely expected to be false. We look at the average growth and prove some new bounds.

May 14, 2019

Sophie Morel (Princeton University):

Abstract: The calculation of the (intersection) cohomology of a Siegel modular variety includes many difficult character identities (the fundamental lemma, for example). In this lecture, I want to concentrate on the character identity appearing at the infinite place, which involves, among other things, stable discrete series characters and appears to be related in some non-obvious way to an identity of Goresky, Kottwitz and MacPherson. Once we strip away all the Lie group complications, our identity becomes a very elementary statement and can proved directly using the geometry of the Coxeter complex of the symmetric group. The relation with the Goresky-Kottwitz-MacPherson identity also becomes clearer; in particular, neither identity follows from the other, but they should have a common generalization. This is joint work with Richard Ehrenborg and Margaret Readdy.

May 14, 2019

Peter Sarnak (Princeton University):

The topology of a real hypersurface in P^n(R) of high degree can be very complicated. However, if we choose the surface at random, there is a universal law. Little is known about this law and it appears to be dramatically different for n=2 and n>2. There is a similar theory for zero sets of monochromatic waves which model nodal sets of eigenfunctions of quantizations of chaotic systems, and in particular of automorphic Maass forms. Joint work with Y. Canzani and I. Wigman.

Academic year 2017-18

September 26, 2017

Sug Woo Shin (UC Berkeley), "Irreducibility of leaves in Shimura varieties"

Abstract: Oort defined central leaves in the special fiber of Shimura varieties as the locus on which the isomorphism class of the universal p-divisible group is constant (when Shimura varieties parametrize abelian varieties with additional structure). Then Chai and Oort proved irreducibility of leaves for Siegel Shimura varieties by geometric methods. In this talk I report on an ongoing work with Arno Kret on proving the irreducibility for Hodge-type Shimura varieties via a different approach using more automorphic input.

September 26, 2017

Xinyi Yuan (UC Berkeley), "Height formulas on Shimura curves"

Abstract: The goal of this talk is to summarize three different height formulas on Shimura curves for quaternion algebras over totally real fields: the Gross--Zagier formula, the formula for the height of a CM point, and the formula for the modular height of the Shimura curve. While the heights and the L-functions involved in different formulas are very different, the proofs of the formulas lie in the same framework.

October 24, 2017

Pierre Colmez (CNRS, Université Pierre et Marie Curie, Paris), "p-adic étale cohomology of the Drinfeld tower and p-adic local Langlands correspondence"

Abstract: It is now classical that the l-adic étale cohomology of the Drinfeld tower, for l not p, encodes both the local Langlands and Jacquet-Langlands correspondences. I will explain that, in dimension 1, the p-adic étale cohomology of this tower encodes part of the p-adic local Langlands correspondence (this is joint work with Gabriel Dospinescu and Wieslawa Niziol).

October 24, 2017

Wieslawa Niziol (CNRS, École Normale Supérieure de Lyon), "Cohomology of p-adic Stein spaces"

Abstract: I will discuss a comparison theorem that allows us to recover p-adic (pro-)étale cohomology of p-adic Stein spaces with semistable reduction over local rings of mixed characteristic from complexes of differential forms. Present applications include a computation of the p-adic étale cohomology of the Drinfeld half-space in any dimension and of its coverings in dimension 1. This is a joint work with Pierre Colmez and Gabriel Dospinescu.

December 5, 2017

A. Raghuram (IISER Pune), "Arithmetic properties of automorphic L-functions"

Abstract: This talk will be an exposition of a circle of ideas that concerns the cohomology of arithmetic groups and the special values of automorphic L-functions. I will begin by introducing the general context in which one can study the notion of Eisenstein cohomology. I will then explain some results of Harder on the cohomology of the boundary of the Borel-Serre compactification of a locally symmetric space and it's relation with induced representations of the ambient reductive group. Once this context is in place one may then try to view Langlands's constant term theorem, which sees the ratios of products of automorphic L-functions, in terms of maps in cohomology. Whenever this is possible one is able to prove rationality results for ratios of critical values of certain automorphic L-functions. I will explain some recent results about (1) Rankin-Selberg L-functions of GL(n) x GL(m)--mostly in collaboration with Günter Harder, and (2) L-functions for SO(n,n) x GL(1)--in collaboration with Chandrasheel Bhagwat.

December 5, 2017

Adrian Iovita (Concordia), "Triple product p-adic L-functions for finite slope p-adic families of elliptic modular forms"

Abstract: In recent joint work with F. Andreatta we constructed modular sheaves of arbitrary weights on strict neighbourhoods of the ordinary locus of modular curves interpolating the symmetric powers of the relative de Rham cohomology sheaves of the universal generalized elliptic curve, endowed with natural filtrations and connections. One of the applications of these constructions is the definition of the triple product p-adic L-functions attached to three finite slope p-adic families of modular forms, extending previous definitions of Hida and Darmon-Rotger for ordinary families.

February 13, 2018

Erez Lapid (Weizmann Institute of Science and IAS), "Parabolic induction of representations of the general linear group over a non-archimedean local field and geometry"

Abstract: Determining irreducibility of parabolic induction is important for the analysis of the automorphic discrete spectrum, as well as in its own right. I will present some results and conjectures about the case of the general linear group, which goes back to the seminal work of Joseph Bernstein and Andrei Zelevinsky in the 1970s. In particular, we prove a special case of a (strong form of a) conjecture of Geiss-Leclerc-Schröer. This is joint work with Alberto Mínguez. I will also discuss an apparently new conjecture about certain parabolic Kazhdan-Lusztig polynomials. We verified this conjecture numerically in a small range by computing all Kazhdan-Lusztig polynomials for the symmetric group S(12).

February 13, 2018

George Pappas (Michigan State), "Good and semi-stable reduction of Shimura varieties"

Abstract: We will describe a classification of Shimura varieties that have good or semi-stable reduction at a prime where the level subgroup is parahoric. This is joint work with X. He and M. Rapoport.

March 13, 2018

Manish Patnaik (U. Alberta), "Automorphic forms on (metaplectic covers of) Kac-Moody groups"

Abstract: Both the Langlands-Shahidi method for studying automorphic L-functions and the Weyl group multiple Dirichlet series approach to studying moments of L-functons have conjectural extensions to infinite-dimensional Kac-Moody groups. We will explain some recent progress in these areas, especially in the function field setting and for (metaplectic covers of) affine Kac-Moody groups.

March 13, 2018

Kumar Murty (U. Toronto), Euler-Kronecker constants

Abstract: Ihara defined, and began the systematic study of, the Euler-Kronecker constant of a number field. In some cases, these constants arise in the study of periods of Abelian varieties. For abelian number fields, they can be explicitly connected to subtle problems about the distribution of primes. In this talk, we review some known results and describe some joint work with Mariam Mourtada.

May 8, 2018

Ellen Eischen (U. Oregon), "p-adic L-functions"

Abstract: : I will discuss a construction of p-adic L-functions, with a focus on the setting of unitary groups. I will highlight how this construction connects to more familiar ones of Serre, Katz, and Hida, and I will emphasize the role of properties of certain automorphic forms (analogous to the role played by modular forms in their work). This includes joint work with Michael Harris, Jian-Shu Li, and Christopher Skinner.

May 8, 2018

Arul Shankar (U. Toronto), "Polynomials with Squarefree Discriminant"

Abstract: A classical question in analytic number theory is understanding the density of squarefree values taken by an integer polynomial. In this talk, we will consider a special class of polynomials, namely, discriminant polynomials. In this case, we use methods from arithmetic invariant theory in conjunction with analytic methods to demonstrate that a positive proportion of integer polynomials (of fixed degree) have squarefree discriminant. This is joint work with Manjul Bhargava and Xiaoheng Wang.

Academic year 2016-17

September 13, 2016

Raf Cluckers (Université de Lille I)

September 13, 2016

Baiying Liu (Purdue University), "On the local converse theorem for p-adic GL_n"

Abstract: In this talk, I will introduce a complete proof of a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field. This is a joint work with Prof. Herve Jacquet. (http://arxiv.org/abs/1601.03656)

October 18, 2016

Chantal David (Concordia University), "One-parameter families of elliptic curves with non-zero average root number"

Abstract: We investigate in this talk the average root number of one-parameter families of elliptic curves (ie. elliptic curves over Q(t), or elliptic surfaces over Q). Helfgott showed that, under Chowla's conjecture and the square-free conjecture, the average root number is 0 unless the curve has no place of multiplicative reduction over Q(t). We first classify elliptic surfaces y^2=x^3+a_2(t)x^2+a_4(t)x+a_6(t) with a_i(t)\in Z[t] and deg(a_i(t)) \le 2 and no place of multiplicative reduction, and compute the average root number for some of the families. Some families have periodic root number, giving a rational average, and some other families have an average root number which is expressed as an infinite Euler product.

We then use those constructed families to exhibit non-isotrivial families of elliptic curves with excess rank, i.e. rank r over Q(t) and average root number -(-1)^r. We also show that the average root numbers (over Z and over Q) over some of those (non-isotrivial) families are dense in [-1,1].

This is joint work with S. Bettin and C. Delaunay

October 18, 2016

Ali Altug (MIT), "On the interaction between the discrete and geometric parts of the trace formula"

Abstract: Let G be a semisimple group over a number field F. A central problem in the theory of automorphic forms and number theory is to understand the structure of the discrete part of the space L^2(G(F)\G(A_F)). The celebrated Arthur-Selberg trace formula contains all the information about the discrete spectrum, however to extract this information for application one needs to isolate the contribution of the discrete spectrum in the so-called "discrete part" of the trace formula, which consists of all the distributions that occur discretely in the trace formula (there are more terms in the discrete part than just the trace of the operator on the discrete spectrum!). In this talk I will introduce the problem and give some motivation. Then talk about what is known and what may be within reach, and introduce further problems suggested recently by Arthur.

November 15, 2016

Yuri Zarhin (Penn State), "Division by 2 on hyperelliptic curves and jacobians"

Abstract: Let $g>1$ be an integer. Suppose that $C$ is a genus $g$ hyperelliptic curve that is canonically embedded into its $g$-dimensional jacobian $J$ in such a way that one of the Weierstrass points goes to zero. For each ``finite" point $P$ of $C$ we describe explicitly the Mumford representations of all $2^{2g}$ halves of $P$ in $J$. As an application, we prove that the genus 2 curve $y^2=x^5-x+1$ does not contain points of odd order >1.

November 15, 2016

Luca Candelori (Louisiana State University), "The transformation laws of algebraic theta functions"

Abstract: We present the algebro-geometric theory underlying the classical transformation laws of theta functions with respect to the action of symplectic matrices on Siegel's upper half-space. More precisely, we explain how the theta multiplier, the half-integral weight automorphy factor and the Weil representation occuring in the classical transformation laws all have a geometric origin, that is, they can all be constructed within a given moduli problem on abelian schemes. To do so, we introduce and study new algebro-geometic constructions such as theta multiplier bundles, metaplectic stacks and bundles of half-forms, which could be of independent interest. Applications include a geometic theory of modular forms of half-integral (in the sense of Shimura), and their generalizations to higher degree, as well as giving new, explicit formulas for determinant bundles on abelian schemes.

February 14, 2017

Philippe Michel (École Polytechnique Fédérale de Lausanne), "The second moment of central value of twisted L-functions: proof and applications"

In a series of recent works Blomer, Fouvry, Kowalski, Milicevic, Sawin and myself have been able to solve the vexing problem of evaluating asymptotically the second moment of the central L-values of character twists (of large prime conductor) of a fixed modular form; the solution combine the spectral theory of modular forms, bounds for bilinear sums of Kloosterman sums and advanced methods in l-adic cohomology. We will describe this and some recent applications (by the same authors) to the non-vanishing of these central values.

February 14, 2017

Lillian Pierce (Duke University), "p-torsion in class groups of number fields of arbitrary degree"

Abstract: Fix a number field K of degree n over the rationals, and a prime p, and consider the p-torsion subgroup of the class group of K. How big is it? It is conjectured that this p-torsion subgroup should be very small (in an appropriate sense), relative to the absolute discriminant of the field; this relates to the Cohen-Lenstra heuristics and various other arithmetic problems. So far it has proved extremely difficult even to beat the trivial bound, that is, to show that the p-torsion subgroup is noticeably smaller than the full class group. In 2007, Ellenberg and Venkatesh shaved a power off the trivial bound by assuming GRH. This talk will discuss several new, contrasting, methods that recover this bound for almost all members of certain families of fields, without assuming GRH. This includes recent joint work with Jordan Ellenberg, Melanie Matchett Wood, and Caroline Turnage-Butterbaugh.

March 21, 2017

Tim Browning (University of Bristol), "Rational curves on smooth hypersurfaces of low degree"

I will discuss recent joint work with Pankaj Vishe, in which we are able to say something about the naive moduli space of rational curves on arbitrary smooth hypersurfaces of sufficiently low degree, by invoking methods from analytic number theory.

March 21, 2017

Roger Heath-Brown (University of Oxford), "Gaps between smooth numbers"

There are many reasons for studying smooth numbers. They provide a toy example, helpful for understanding primes. We will look at the mean square difference between consecutive smooth numbers. The investigation leads to a novel mean value problem for Dirichlet polynomials.

April 4, 2017

James Maynard (University of Oxford), "Polynomials representing primes"

Abstract: It is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. Unfortunately, this isn't known in any case except for linear polynomials - the sparsity of values of higher degree polynomials causes substantial difficulties. If we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on `few' values; Friedlander-Iwaniec showed X2+Y4 is prime infinitely often, and Heath-Brown showed the same for X3+2Y3. We will demonstrate a family of multivariate sparse polynomials all of which take infinitely many prime values.

April 4, 2017

Kartik Prasanna (University of Michigan) "Hodge classes and the Jacquet-Langlands correspondence"

Abstract: I will discuss the relation between Langlands functoriality and the theory of algebraic cycles in one of the simplest instances of functoriality, namely the Jacquet-Langlands correspondence for Hilbert modular forms. In this case, the Tate conjecture predicts that functoriality is realized by an algebraic cycle. While we cannot yet show the existence of such a cycle, I will outline an unconditional proof of the existence of the corresponding Hodge class and give some applications. This is joint work (in progress) with A. Ichino

Academic year 2015-16

September 15, 2015

Carl Pomerance (Dartmouth College), "The sum-of-proper-divisors function"

Abstract: Introduced by Pythagoras, the sum-of-proper-divisors function may be the very first function of mathematics. Its study spurred the development of elementary number theory and the field of probabilistic number theory among much else. Pythagoras suggested iterating this function (so, perhaps the first dynamical system!), finding the 2-cycle 220 and 284. This talk will discuss some very recent results on the range of the sum-of-proper-divisors function, the distribution of 2-cycles, and some related problems. Co-authors on various aspects of this work include Florian Luca and Paul Pollack.

September 15, 2015

Jennifer Balakrishnan (University of Oxford), "Variations on quadratic Chabauty"

Abstract: Let C be a curve over the rationals of genus g at least 2. By Faltings' theorem, we know that C has finitely many rational points. When the Mordell-Weil rank of the Jacobian of C is less than g, the Chabauty-Coleman method can often be used to find these rational points through the construction of certain p-adic integrals.

When the rank is equal to g, we can use the theory of p-adic height pairings to produce p-adic double integrals that allow us to find integral points on curves. In particular, I will discuss how to carry out this ``quadratic Chabauty'' method on hyperelliptic curves over number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).

October 6, 2015

John Cremona (University of Warwick), "Black box Galois representations"

October 6, 2015

Stefan Wewers (Ulm University), "Explicit Chabauty-Kim theory for the thrice punctured line and mixed Tate motives"

Abstract: Chabauty-Kim theory tries to prove finiteness of integral points on hyperbolic curves (i.e. the theorem of Siegel and Faltings) using the motivic fundamental group. I will report on our project to make this theory explicit in the case of the thrice punctured line. This leads us to the problem of explictily representing mixed Tate motives as motivic polylogarithms. Joint work with Ishai Dan-Cohen.

November 3, 2015

Ralf Schmidt (University of Oklahoma), "Newforms for GSp(4) and the metaplectic group"

Abstract: The classical Atkin-Lehner theory of new- and oldforms is well known. In this talk we will present a similar theory for Siegel modular forms of degree 2 with respect to the paramodular group. This theory is based on the local representation theory of the group GSp(4). Local representation theory can also explain why an analogous newform theory for the more familiar Siegel congruence subgroups does NOT exist. Progress on Siegel congruence subgroups can be made via a connection to the metaplectic group, the double cover of SL(2). We will present some results on the metaplectic group and their consequences for GSp(4). This is joint work with Brooks Roberts.

November 3, 2015

Henri Cohen (Universite Bordeaux I), "Numerical algorithms in number theory"

Abstract: The goal of the talk is to present algorithms (all available in Pari/GP or Sage) of numerical (as opposed to arithmetical) nature useful in number theory, such as multiprecision numerical integration, summation, extrapolation, multiple zeta values and polylogs, inverse Mellin transforms, and numerical computation of L-functions. (Slides for the talk)

February 9, 2016

David Soudry (Tel Aviv University), "On Rankin-Selberg integrals for classical groups"

Abstract: I will survey the structure of families of global integrals of Rankin-Selberg type, which were predicted to represent partial L-functions for pairs of irreducible, automorphic, cuspidal representations (\pi, \tau) on (G, GL_n), where G is a classical group. I will focus on split orthogonal groups. In the global integrals, we integrate a Fourier coefficient "of Bessel type" applied to a cusp form on G against an Eisenstein series on a related classical group H, induced from a maximal parabolic subgroup, or vice versa. These families of integrals contain all known ones which represent the partial L-functions above. They were first introduced by Ginzburg, Piatetski-Shapiro and Rallis, and were calculated in the so-called "spherical case" (of Bessel models). I will present the calculation of the unramified local integrals at all cases. It is done by "analytic continuation" from the generic cases above (which were known long before). The global integrals above are useful in locating poles of L-functions of representations \pi with a given type of Bessel models.

February 9, 2016

Bhargav Bhatt (University of Michigan), "Integral p-adic Hodge theory"

Abstract: I will describe a new cohomology theory for a proper smooth scheme over a p-adic ring. This theory interpolates between the existing ones (etale, de Rham, crystalline), and sheds some new light on the behavior of torsion in cohomology as an algebraic variety degenerates from characteristic 0 to characteristic p. This talk is based on joint work with M. Morrow and P. Scholze.

March 1, 2016

David Zywina (Cornell), "Possible indices for the Galois image of elliptic curves over Q"

Abstract: For a non-CM elliptic curve E/Q, its Galois action on all its torsion points can be expressed in terms of a Galois representation. A famous theorem of Serre says that the image of this representation is as "large as possible" up to finite index. We will study what indices are possible assuming that we are willing to exclude a finite number of possible j-invariants from consideration.

March 1, 2016

Keerthi Madapusi Pera (University of Chicago), "On the average height of abelian varieties with complex multiplication"

Abstract: In the 90s, generalizing the classical Chowla-Selberg formula, P. Colmez formulated a conjectural formula for the Faltings heights of abelian varieties with multiplication by the ring of integers in a CM field, which expresses them in terms of logarithmic derivatives at 1 of certain Artin L-functions. Using ideas of Gross, he also proved his conjecture for abelian CM extensions. In this talk, I will explain a proof of Colmez's conjecture in the average for an arbitrary CM field. This is joint work with F. Andreatta, E. Goren and B. Howard.

April 12, 2016

David Savitt (Johns Hopkins University), "Moduli stacks of potentially Barsotti-Tate Galois representations"

Abstract: I will discuss joint work with Ana Caraiani, Matthew Emerton, and Toby Gee in which we construct moduli stacks of two-dimensional tamely potentially Barsotti-Tate Galois representations, and relate their geometry to the weight part of Serre's conjecture for GL(2).

April 12, 2016

Armand Brumer (Fordham University), "On the paramodular conjecture"

Abstract: After reviewing what is known about modularity for abelian surfaces, we'll focus on two results: i) A uniqueness criterion for the isogeny class of certain abelian surfaces. ii) How Serre's 1984 "quartic method", used to check modularity of some elliptic curves, can be adapted to do the same for certain abelian surfaces. This is joint work with Ken Kramer, Cris Poor, David Yuen and John Voight.