# BC-MIT number theory seminar abstracts

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

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), TBA

Abstract: TBA

February 13, 2018
George Pappas (Michigan State), TBA

Abstract: TBA

March 13, 2018
Manish Patnaik (U. Alberta), TBA

Abstract: TBA

March 13, 2018
Kumar Murty (U. Toronto), TBA

Abstract: TBA

May 8, 2018
Ellen Eischen (U. Oregon), TBA

Abstract: TBA

May 8, 2018
Arul Shankar (U. Toronto), TBA

Abstract: TBA

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

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.

Schedules of yesteryear.