Seminars Sorted by Series

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...

The families of motives of the title arise from classical one-variable hypergeometric functions. This talk will focus on the calculation of their corresponding L-functions both in theory and in practice. These L-functions provide a fairly wide class...

This talk will be about some new tools based on model theory that could be useful in the study of harmonic analysis on reductive p-adic groups. In 2008, R. Cluckers and F. Loeser defined the class of the so-called "constructible motivic exponential...

I will report some recent progress in the study of local models of Shimura varieties, including the proof of the coherence conjecture of Pappas-Rapoport and the Kottwitz conjecture

I will report on some recent work on multiple zeta values. I will sketch the definition of motivic multiple zeta values, which can be viewed as a prototype of a Galois theory for certain transcendental numbers, and then explain how they were used to...

1. A (very) special case of Deligne-Lusztig theory yields a construction of cuspidal irreducible representations of the finite group $GL_n(\mathbb F_q)$ in the cohomology of an algebraic variety equipped with an action of $GL_n(\mathbb F_q)$. There...

I will describe the appearance of special values of Eisenstein series on E6, E7, and E8 that arose in studying the low energy expansion of the 4-graviton scattering amplitude in string theory (see arxiv:1004.0163 and 1111.2983). Through methods to...

For GL(2) over Q_p, the p-adic Langlands correspondence is available in its full glory, and has had astounding applications to Fontaine-Mazur, for instance. In higher rank, not much is known. Breuil and Schneider put forward a conjecture, which is a...

We'll prove a converse theorem for forms forms on SL_2. While the theorem is easy to prove once it has been formulated, the number-theoretic considerations leading to its formulation nevertheless pose some interesting and apparently unsolved...

Recently Nicolas and Serre have determined the structure of the Hecke algebra acting on modular forms of level 1 modulo 2, and Serre has conjectured the existence of a universal Galois representation over this algebra. I'll explain the proof of this...

In his work of the early 1980s, Shimura observed that expressions of special values of automorphic L-functions in terms of period invariants could be used to obtain relations among the latter. This observation has since been applied in numerous...

The classical Kuga-Satake construction, over the complex numbers, uses Hodge theory to attach to each polarized K3 surface an abelian variety in a natural way. Deligne and Andre extended this to fields of characteristic zero, and their results can...

To any essentially self-dual, regular algebraic (ie cohomological) automorphic representation of GL(n) over a CM field one knows how to associate a compatible system of l-adic representations. These l-adic representations occur (perhaps slightly...

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. Under the generalized Sato-Tate conjecture, this is equal to the distribution of characteristic polynomials of...

Monodromy groups arise naturally in algebraic geometry and in differential equations, and often preserve an integral lattice. It is of interest to know whether the monodromy groups are arithmetic or thin. In this talk we review the Deligne-Mostow...

We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. Motivated by work of Kato and others for n=3, we show that local-global principles hold for H^n(F, Z/mZ(n-1)) for all n>1...

Abstract: Associated to an abelian variety A/K is a Galois representation which describes the action of the absolute Galois group of K on the torsion points of A. In this talk, we shall describe how large the image of this representation can be (in...

Application of Plancherel's theorem to integral kernels approximating compact period functionals yields estimates on (global) automorphic Levi-Sobolev norms of the functionals. The utility of this viewpoint can be illustrated in reconsideration of...

Let $X$ be a locally symmetric variety, $\bar{X}$ its Baily-Borel compactification, $\bar{X}^{rbs}$ its reductive Borel-Serre compactification and $p:\bar{X}^{rbs} \to \bar{X}$ the canonical map. We prove that the derived direct image sheaf $Rp_*...

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on GL(n+1) x GL(n) along the diagonal subgroup GL(n) and express it in terms of the Rankin-Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This...