# Seminars & Events for Princeton University/IAS Number Theory Seminar

##### Towards weak p-adic Langlands for GL(n)

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 somewhat coarse version of p-adic Langlands for GL(n). It roughly says that an admissible filtration exists if and only if an invariant norm exists. One direction (producing a filtration) was completely proved by Hu in his Orsay thesis. In my talk, I will report on progress in the opposite direction (producing a norm). I will first sketch the proof in the "indecomposable" case, where one can pass to a global setting by a standard trace formula argument, and construct a norm from p-adic modular forms.

##### A Converse Theorem for SL_2

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 questions.

##### Modular forms modulo 2

##### On the parity of coefficients of modular forms.

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 conjecture, and show how that representation may be used to get new information on the parity of the coefficients of modular forms of level 1 -- for example, on the parity of the values of the generalized Ramanujan's tau functions. I'll also explain a conjectural relation with the partition function.

##### Central values of Rankin-Selberg L-functions and period relations

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 situations by the speaker, among others. Most of these applications involve the theta correspondence. This talk will describe a different approach, based on the Ichino-Ikeda

##### The Tate conjecture for K3 surfaces over fields of odd characteristic

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 be combined with Faltings's isogeny theorem to prove the Tate conjecture for K3 surfaces in characteristic zero. Using the theory of integral canonical models of Shimura varieties of orthogonal type, we extend the Kuga-Satake construction to odd characteristic. We can then deduce the Tate conjecture for K3s in this situation as well (with some exceptions in characteristic 3).

##### Galois representations for regular algebraic cuspidal automorphic forms

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 twisted) in the cohomology of a Shimura variety. Recently Harris, Lan, Thorne and myself have constructed l-adic representations without the `essentially self-dual' hypothesis'. In this case the l-adic representations do not occur in the cohomology of any Shimura variety. Rather we construct them using a congruence argument. In this talk I will describe this theorem and sketch the proof.

##### Sato-Tate distributions in genus 2

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 random matrices in a closed subgroup ST(A) of USp(4). The Sato-Tate group ST(A) may be defined in terms of the Galois action on any Tate module of A, and must satisfy a certain set of constraints (the Sato-Tate axioms). Up to conjugacy, we find that there are exactly 55 subgroups of USp(4) that satisfy these axioms. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Monodromy and arithmetic groups

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 theory and show that for cyclic coverings of degree $d$ of the projective line, with a prescribed number $m$ of branch points and prescribed ramification indices, the monodromy is an arithmetic group provided $m\geq 2d-2$. PLEASE CLICK ON THE SEMINAR TITLE FOR THE FULL ABSTRACT.

##### Local Global Principles for Galois Cohomology

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 $H^n(F, {\mathbb Z}/m{\mathbb Z} (n-1))$ for all $n>1$. In the case $n=1$, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for $H^1(F,G)$, a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups $G$. Concrete applications of the results include central simple algebras and Albert algebras.

##### Counting Rational Points on Cubic Surfaces

##### Abelian varieties with maximal Galois action on their torsion points

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 terms of a number field K and the dimension of A). We achieve this by considering abelian varieties in families and then using a special variant of Hilbert's irreducibility theorem. Some results of Serre on the mod ell Galois image will also be reviewed. (This is joint work with David Zureick-Brown)

##### Automorphic Levi-Sobolev Spaces, Boundary-Value Problems, and Self-Adjoint Operators

Application of Plancherel's theorem to integral kernels approximating compact period functionalsyields estimates on (global) automorphic Levi-Sobolev norms of the functionals. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### An introduction to motives

**PLEASE NOTE SPECIAL TIME AND LOCATION.** ** This is in addition to the regular PU/IAS Number Theory Seminar**. We review the construction of the triangulated categories of motives over a base scheme (following the method of Morel and Voevodsky). We then explain quickly the construction of various operations between these categories as well as some realization functors (Betti and $\ell$-adic). If time permits, we also discuss the rigid analytic variant and develop an application to the theory of nearby and vanishing cycles.

##### Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety

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 $Rp_*\mathbb{Q}$ is the realization of a canonical motive associated to the variety $\bar{X}$. This is non trivial since $\bar{X}^{rbs}$ is not an algebraic variety in general. (Joint with S. Zucker)

##### Regularized periods of automorphic forms

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. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Compactifications of PEL-type Shimura varieties and Kuga families with ordinary loci

I will report on the construction of p-integral models of various algebraic compactifications of PEL-type Shimura varieties and Kuga families, allowing ramification (including deep levels) at p, with good behaviors over the loci where certain (multiplicative) ordinary level structures are defined. (We know almost nothing about the non-ordinary loci when p is ramified, but this theory is still sufficient for at least one important application.) This is a somewhat notorious subject, but I will try to begin with motivations and qualitative descriptions that might (it is hoped) make things easier.

##### Standard and nonstandard comparisons of relative trace formulas

The trace formula has been the most powerful and mainstream tool in automorphic forms for proving instances of Langlands functoriality, including character relations. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Goren-Oort stratification of Hilbert modular varieties mod p and Tate conjecture

In this talk, I will report on an on-going joint project with David Helm and Yichao Tian. Let p be a prime unramified in a totally real field F. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### An analogue of the Ichino-Ikeda conjecture for Whittaker coefficients of the metaplectic group

A few years ago Ichino-Ikeda formulated a quantitative version of the Gross-Prasad conjecture, modeled after the classical work of Waldspurger. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.