Seminars Sorted by Series

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...

An interesting new class of modular forms has emerged in the last several years that generalize Ramanujan's mock theta functions. The generalization is based on an observation of of Zwegers who showed that mock theta functions occur as holomorphic...

The automorphic cohomology of a reductive $\mathbb{Q}$-group $G$, defined in terms of the automorphic spectrum of $G$, captures essential analytic aspects of the arithmetic subgroups of $G$ and their cohomology. We discuss the actual construction of...

We will give a geometric definition of the notion of overconvergent modular form of any p-adic weight. As a consequence, we re-obtain Coleman's theory of p-adic families of eigenforms and the eigencurve of Coleman and Mazur without using the...

A rational elliptic curve may be viewed as the set of solutions to an equation of the form $y^2=x^3+Ax+B$, where $A$ and $B$ are rational numbers. It is known that the rational points on this curve possess a natural abelian group structure, and the...

Write $a \prec b$ if $b \equiv 1 \pmod{a}$. A prime chain is a chain $p_1 \prec p_2 \prec \dotsb \prec p_k$ whose components are prime numbers. In my talk, I will sketch the proof of the fact that the number of prime chains above $p$ (i.e., with $p...

We fix F a local non archmedean field of characteristic zero. Let G the points over F of an algebraic reductive group defined over F and s a rational involution of G defined over F. We denote by H the group of fixed points of G under the action of s...

To an abelian variety over a number field one can associate an abelian variety to each prime ideal p of good reduction by reducing the variety modulo p . The geometry of these reductions need not resemble the geometry of the original abelian variety...

The motivating open question for this talk is to find for a given prime p all local Z_p-algebras which can occur as the deformation ring of some linear representation over F_p of some finite group G. We will show for every p that not all of them are...

The motivating open question for this talk is to find for a given prime p all local Z_p-algebras which can occur as the deformation ring of some linear representation over F_p of some finite group G. We will show for every p that not all of them are...

In joint work with Barry Mazur, we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions for an elliptic curve to have twists of arbitrary 2-Selmer rank, and we...

In 1993, Sczech defined an n-1 cocycle on GL_n(Z) valued in a certain space of distributions. He showed that specializations of this cocyle yield the values of the partial zeta functions of totally real fields of degree n at nonpositive integers. In...

Fontaine and Mazur have a remarkable conjecture that predicts which (p-adic) Galois representations arise from geometry. In the special case of two dimensional representations with distinct Hodge-Tate weights, they further conjecture that these...

If K is the rational function field K=Q(t), then a polynomial f in K[x] can be regarded as a one-parameter family of polynomials. If f is irreducible, then a basic form of Hilbert's irreducibility theorem states that there are infinitely many...

We discuss the following question of Nick Katz and Frans Oort: Given an Algebraically closed field K , is there an Abelian variety over K of dimension g which is not isogenous to a Jacobian? For K the complex numbers its easy to see that the answer...

In this talk, I will state some conjectures and examples concerning the central derivatives of automorphic L-series in terms of heights of special cycles on Shimura varieties.

In a recent paper, DeBacker and Reeder have constructed a piece of the local Langlands correspondence for pure inner forms of unramified p-adic groups and have shown that the corresponding L-packets are stable. In this talk we are going to discuss...

We will describe how the "special value theory" in function field arithmetic is an interesting mixture of very strong theorems determining all algebraic relations in some cases, emerging partial conjectural pictures in some cases, and quite wild...

This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this...

It is known that the roots of congruences for a fixed irreducible quadratic polynomial are equidistributed. This statement translates to getting cancellation in the corresponding sum of Weyl's sums. In a recent work by W. Duke, J. Friedlander and H...

We proved with Umberto Zannier that there are at most finitely many complex numbers $\lambda \neq 0,1$ such that two points on the Legendre elliptic curve $y^2=x(x-1)(x-\lambda)$ with coordinates $x=2$ and $x=3$ both have finite order. However we...

The $\lambda$-invariant is an invariant of an imaginary quadratic field that measures the growth of class numbers in cyclotomic towers over the field. It also measures the number of zeroes of an associated $p$-adic L-function. In this talk, I will...