Seminars Sorted by Series

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...

It is well-known that there are connections between the representation theory of a reductive p-adic group and the topology of the flag variety of the Langlands L-group. We will discuss Whittaker functions in this light. The Casselman-Shalika formula...

In the early 80's, Shimura made a precise conjecture relating Petersson inner products of arithmetic automorphic forms on quaternion algebras over totally real fields, up to algebraic factors. This conjecture (which is a consequence of the Tate...

I will talk about the fundamental theorem of affine sieve (joint with Sarnak). The main black box in the proof of this result will be also explained. It is a theorem on a necessary and sufficient condition for a finitely generated subgroup of SL(n,Q...

Among the bounty of brilliancies bequeathed to humanity by Srinivasa Ramanujan, the circle method and the notion of mock theta functions strike wonder and spark intrigue in number theorists fresh and seasoned alike. The former creation was honed to...

We show that the p-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over F_p. By modeling this intersection as the intersection of a random...

I will discuss some possible applications of algebraic cycles and p-adic families of modular forms to the arithmetic of elliptic curves over abelian extensions of real quadratic fields. This is a report on work in progress with Victor Rotger and...

We give a survey of recent results on conjectures of Heath-Brown and Serre on the asymptotic density of rational points of bounded height. The main tool in the proofs is a new global determinant method inspired by the local real and p-adic...

In 1969 Artin and Mazur defined the etale homotopy type Et(X) of scheme X, as a way to homotopically realize the etale topos of a X. In the talk I shall present for a map of schemes X --> S a relative version of this notion. We denoted this...

By a celebrated theorem of Hurwitz, a curve X/C of genus g>1 has at most 84(g-1) points. Curves that attain or come close to this bound, such as the modular curves X(N) , often have a rich structure with diverse connections to and near number theory...

A. Ghosh and P. Sarnak have recently initiated the study of so-called real zeros of holomorphic Hecke cusp forms, that is zeros on certain geodesic segments on which the cusp form (or a multiple of it) takes real values. In the talk I'll first...

Rankin-Selberg integrals, among many other things they do, are the only way to prove that special values $L(f,n)$ of L-functions of weight $k$ modular forms on $GL_2(\mathbb Q)$, $n \geq k$, are periods. They pave the road to Beilinson’s motivic $...

In this talk, I will present a formulation of the Gross-Zagier formula over Shimura curves using automorphic representations with algebraic coefficients. It is a joint work with Shou-wu Zhang and Wei Zhang.

Let D be a p-divisible group over an algebraically closed field k of positive characteristic p. We will first define several subtle invariants of D which have been introduced recently and which are crucial for any strong, refined classification of D...

I will introduce an arithmetic version of the classical Rallis' inner product formula for unitary groups, which generalizes the previous works of Kudla, Kudla-Rapoport-Yang and Bruinier-Yang. As Rallis' formula concerns the central L-values of...

We prove the existence of second main term of order X^{5/6} for the function counting cubic fields. This confirms a conjecture of Datskovsky-Wright and Roberts. We also prove a variety of generalizations, including to arithmetic progressions, where...

We consider a generalization of Serre's conjecture for irreducible, conjugate self-dual Galois representations rho : G_F --> GL_3(\bar F_p), where F is an imaginary quadratic field in which p splits. We previously gave a conjecture for the possible...

This talk is based on some results for real groups used in Arthur's classification of global packets for classical groups. The setting is twisted endoscopy for a connected reductive algebraic group over the reals. There is a geometric transfer which...

In this talk we consider the problem of giving explicit inequalities which relate heights, discriminants and conductors of a curve defined over a number field. We present such inequalities for some curves, including all curves of genus one or two...

In this talk we will show that many arithmetic sequences have asymmetries in their distribution amongst the progressions mod q. The general phenomenon is that if we fix an integer a having some arithmetic properties (these properties depend on the...

We will discuss the value distribution of the Epstein zeta function E_n(L,cn) for real c and a random lattice L of covolume 1 and large dimension n. Important ingredients in our study will be the distribution of vector lengths in a random lattice as...

We begin with a survey of recent results on the problem of bounding the sup-norm of automorphic forms. If f is a cuspidal automorphic forms on a reductive group G it is classical to study its value distribution and in the particular the maximum of...

A fake projective plane is a smooth complex projective algebraic surface whose Betti numbers are same as those of the complex projective plane but which is not the complex projective plane. The first fake projective plane was constructed by David...

Modularity lifting theorems were introduced by Taylor and Wiles and formed a key part of the proof of Fermat's Last Theorem. Their method has been generalized successfully by a number of authors but always with the restriction that the Galois...

Let X a curve over F_q and G a semi-simple simply-connected group. The initial observation is that the conjecture of Weil's which says that the volume of the adelic quotient of G with respect to the Tamagawa measure equals 1, is equivalent to the...

A conjecture of Langlands-Rapoport predicts the structure of the mod p points on a Shimura variety. The conjecture forms part of Langlands' program to understand the zeta function of a Shimura variety in terms of automorphic L-functions. I will...