Seminars Sorted by Series

Weyl group multiple Dirichlet series are multiple Dirichlet series in several variables whose coefficients involve Gauss sums and also reflect the combinatorics of a given root system. The earliest examples came from Mellin transforms of metaplectic...

In this talk, I will explain how to compute the trace of a power of the Frobenius endomorphism on the intersection cohomology of the Baily-Borel compactification of a Siegel modular variety. The main tools are : - Kottwitz's calculation of the...

In this talk, I shall introduce a sieve method for bounding the average size of shifted convolution summation terms related to the Quantum Unique Ergodicity Conjecture for a fixed Hecke-Maass cusp form. This bound will be uniform in the spectral...

I describe a new simple way to obtain Rankin-Selberg type spectral identities. These include the classical Rankin-Selberg identity, the Motohashi identity for the forth moment of the zeta function and many new identities between various L-functions...

p-adic height pairings are analogues of the real height pairings that show up for example in the Birch-Swinnerton Dyer conjectures on special values of L-functions of Abelian varieties. The p-adic pairings show up when one tries to formulate p-adic...

A well known theorem of Duke states that the collection of all closed geodesics of a given discriminant on SL(2,Z) \ H becomes equidistributed as the discriminant goes to infinity and can be proved using subconvex estimates for L-functions...

Thanks to the Weil conjectures (proved by Deligne) we know that counting points of varieties over finite fields yields topological information about them. In this talk I will describe such a calculation for certain character varieties...

A conjecture of Lang (on elliptic curves) generalized by Silverman on abelian varieties predicts that the Neron-Tate height of a point on an abelian variety should grow at least like the height of the variety itself. We shall suggest higher...

In this talk I will explain some recent results on local-global compatibility for the action of GL_2 of the finite adeles on the p-adically completed cohomology of modular curves, as well as some applications of these results (e.g. to the Fontaine...

In 1957 Erdos asked what is the smallest maximum modulus of a trigonometric polynomial of degree n all whose coefficients have modulus 1. He thought that there should be a c>0 such that this max modulus is larger than (1+c)sqrt(n).In 1966 Littlewood...

We show that Leopoldt's conjecture for totally real number fields implies Greenberg's conjecture on the uniform boundedness of the $p$-primary part of the class groups of the finite extensions along the cyclotomic tower of a totally real number...

In inverse problems in arithmetic combinatorics, one is interested in describing internal properties of those finite subsets $A$ of an algebraic structure that ``barely expand'' under its operations. One of the deepest results in the area is Freiman...

In this talk I will discuss some recent results on the existence of certain unique models related to the Gross-Prasad conjecture. This is joint with Dipendra Prasad.

In a seminal Inventiones 1976 paper, Hirzebruch and Zagier produced a set of cycles on certain Hilbert modular surfaces whose intersection numbers are the Fourier coefficients of elliptic modular forms with nebentypus. Their result can be viewed as...

A period of an automorphic form on a reductive group G over a number field is defined by its integral over a subgroup H of G. Such periods are often related to special values of automorphic L-functions. In this talk, we present a conjecture in the...

In the beginning we review the classical method of computing the cohomology of PEL-type Shimura varieties with good reduction modulo p, and explain the tools that are useful for studying the bad reduction of compact Shimura varieties. Then we...

For a general L-function, the bound on its critical line obtained by applying the Phragmen-Lindeloff interpolation method is called the convexity bound. Any bounds with a power saving of the convexitybound are called subconvexity bounds. In this...

A sequence of primes p_1, ..., p_k is called a prime chain if p_j | (p_{j+1}-1) for each j ; e.g. 3, 7, 29, 59. We will discuss problems about counting prime chains with certain properties, and about the existence of prime chains with various...

We review the de Branges theory of Hilbert spaces of entire functions. This theory gives a canonical form for a class of operators as a multiplication operator together with a generalized Fourier transform taking such an operator to a generalized...

Studying the Selmer groups of elliptic curves for a supersingular prime is difficult. It turned out we should instead use the plus/minus Selmer groups defined by Kobayashi. In this talk, we will see the plus/minus Selmer group theory for...

For a squarefree integer $d$ we ask, if the negative Pell equation $x^2-dy^2 = -1$ is solvable over the integers. By easy considerations we see that in this case $d>0$ and that all odd prime divisors of $d$ are congruent to 1 modulo 4. Now we call a...

An important result of Waldspurger relates the central value of quadratic base change L-functions for GL(2) to period integrals over tori. Subsequently this result was reproved by Jacquet using the relative trace formula. We will explain some...

The divisor matrix is indexed by the natural numbers with (i,j) entry equal to one if i divides j and 0 otherwise. The convolution of a Dirichlet series with the Riemann zeta function corresponds to multiplication of the sequence of coefficients by...

In a first part, we shall prove a quantitative nonvanishing result conjectured by Ph .Michel and A .Venkatesh which concerns the special derivatives occurring in the Gross-Zagier formula. Our method relies on two classical equidistribution Theorems...

The Mordell-Lang conjecture, proved by Faltings and Vojta, states that a finitely generated subgroup of a semiabelian variety intersects any subvariety of that semiabelian variety in a union of finitely many translates of subgroups. It seems natural...

The subject of partition theory has long been an excellent source of combinatorial hypergeometric q-series that are also automorphic forms. The forms that are associated with even the simplest examples (such as the generating function of the...

Let p be a prime. Let F be an algebraic closure of the finite field F_p with p elements. An integral canonical model N of a Shimura variety Sh(G,X) of Hodge type is a regular, closed subscheme of a suitable pull back of the Mumford moduli tower M...

Suppose that rho is a three-dimensional modular mod p Galois representation whose restriction to the decomposition groups at p is irreducible and generic. If rho is modular in some (Serre) weight, then a representation-theoretic argument shows that...

In this talk, we first describe a systematic way to construct `automorphic Green functions' for Kudla's special divisors on a Shimura variety of orthogonal type $(n, 2)$. We then give an explicit formula for their values at a CM cycle. This formula...

The main motivation for the theory of mock modular forms comes from the desire to provide a framework in which we can understand the mysterious and intriguing mock theta functions, as well as related functions, like Appell functions and theta...

In a couple of recent works with C. Khare and M. Larsen we contruct finte groups of Lie type B_n, C_n and G_2 as Galois groups over rational numbers. The method combines some established, special cases of the functoriality principle with l-adic...

The subconvexity problem consists in providing non-trivial upper bounds for central values of $L$-function. In recent years, this has been recognized as a central point to many arithmetic problems which could be related to the analytic theory of...

Let S be a Shimura variety and L a set of special points on S. Andre and Oort conjecture that any irreducible component of the Zariski-closure of L is a subvariety of Hodge type of S. I will indicate a proof of this conjecture under GRH (this is...