Seminars Sorted by Series

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...

I will describe recent generalizations of mine to a theorem of Harris, Shepherd-Barron, and Taylor, showing that have certain Galois representations become automorphic after one makes a suitably large totally-real extension to the base field. The...

Given an L^2-normalized cusp form f on a modular curve X_0(N) , what can be said about pointwise bounds for f ? For Hecke eigenforms, we will prove the first non-trivial bound in terms of the level N as well as hybrid bounds in terms of the level...

Motivated by the relative trace formula of Jacquet and experience on period integrals of automorphic forms, we take the first steps towards formulating a "relative" Langlands program, i.e. a set of conjectures on H-distinguished representations of a...

We'll discuss the quantum unique ergodicity conjecture of Rudnick and Sarnak and its holomorphic analogue. Highlighting the key ideas in recent joint work with with K. Soundararajan, we'll demonstrate how one may use number theoretic techniques to...

Speaking at a general level, a major goal of the p-adic Langlands program (from a global, rather than local, perspective) is to find a p-adic generalization of the notion of automorphic eigenform, the hope being that every p-adic global Galois...

From the trace formula and the global Langlands correspondence one can infer the existence of a particular rigid l-adic local system on the projective line with tame ramification at 0 and wild ramification, of the mildest possible kind, at infinity...

We will discuss some arithmetic counting problems, ranging from the antique (how many squarefree integers are there in [0 . . N]?) to the au courant (conjectures of Bhargava and Cohen-Lenstra about the distributions of discriminants and of class...

We will explain how toroidal compactifications of certain Kuga families of abelian varieties over integral models of PEL-type Shimura varieties, including for example all those products of universal abelian schemes, can be constructed by a uniform...

In this talk, we will discuss how the circle method can be used to study additive number theory in function fields. After a brief introduction to the circle method and its history, we will describe how the method is implemented in the function field...

We prove, under mild hypotheses, there are no irreducible two-dimensional ordinary even Galois representations of the Galois group of Q with distinct Hodge-Tate weights, in accordance with the Fontaine-Mazur conjecture. We also show how this method...

In this talk I'll present a relative trace formula approach to the Gross-Zagier formula and its high dimensional generalization (a derivative version of the global Gross-Prasad conjecture) for unitary group. In particular, an arithmetic fundamental...

The solution to many classical counting asymptotics problems in number theory goes by comparison with an analogous volume asumptotics. In a general setting, we establish asymptotic formulae for volumes of height balls in analytic varieties over...

I will discuss a recent joint work with Thomas Barnet-Lamb and Toby Gee in which we prove the Sato-Tate conjecture for non-CM regular algebraic cuspidal automorphic representations of GL_2 over a totally real field.

I will discuss a generalization of the modularity lifting theorems of Clozel, Harris and Taylor to the case of ordinary Galois representations. The result is obtained by applying the Taylor-Wiles method (with innovations due to Kisin and Taylor)...

In the 21st-century approach to p-adic Hodge theory, one studies local Galois representations (and related objects) by converting them into modules over certain power series rings carrying certain extra structures (Frobenius actions and derivations)...

We describe how to use Shioda-Inose structures on K3 surfaces to write down explicit equations for Hilbert modular surfaces, which parametrize principally polarized abelian surfaces with real multiplication by the ring of integers in Q({\sqrt{D})...

In the mid 80's Margulis proved the Oppenheim Conjecture regarding values of indefinite quadratic forms. I will present new work, joint with Margulis, where we quantify this statement by giving bounds on the size of integer vectors for which |Q(x)|...

Generalized modular functions are holomorphic functions on the complex upper half-plane, meromorphic at the cusps that satisfy the usual definition of a modular function, however with the important exception that the character need not be unitary...

Suppose E is a rational elliptic curve and p is a given prime. It is of interest to know that there exists a square free integer D such that the D-th quadratic twist E_D of E such that the p-Selmer group of E_D is trivial. The existence of such a D...