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

##### Asymptotics for special derivatives of L-series

In a first part, we shall prove a quantitative nonvanishing result conjectured by Ph.Michel and A.Venkatesh which concerns the special derivatives occuring in the Gross-Zagier formula. Our method relies on two classical equidistribution Theorems in arithmetic geometry. In a second part, we shall explain how to refine these results with methods from analytic number theory. We shall emphasize on the new aspects of this analytic approach which may hopefully be used in other contexts.

##### Gross—Schoen cycles and triple product L-series

##### Dynamical Mordell-Lang problems

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 to ask if such a theorem holds when the finitely generated subgroup is replaced by a finitely generated semigroup of morphisms of a general variety; for example, one might take a semigroup of endomorphisms of a semiabelian variety. We will prove that this is true in many cases when the semigroup is cyclic and also give counterexamples in the more general case, some simple and some more complicated.

##### The coefficients of harmonic Maass forms and combinatorial applications

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 partition function p(n)) have nontrivial levels, characters, and poles at the cusps, and the techniques needed to fully understand the coefficients were long in development. Quite recently, real analytic harmonic Maass forms have also been employed to understand Ramanujan's mysterious mock theta functions and new classes of hypergeometric series.

##### On the Tate and Langlands--Rapoport conjectures for Shimura varieties of Hodge type

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 $M$ over $Z_{(p)}$. We recall that $M$ parametrizes isomorphism classes of principally polarized abelian schemes over $Z_{(p)}$-schemes which have a fixed relative dimension and which have level-$m$ symplectic similitude structures for all $m$ prime to $p$. Deep conjectures of Tate and Langlands--Rapoport pertain to points of $N$ with values in an algebraic closure of the field with $p$ elements.

##### Weight Cycling and Serre-type Conjectures

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 it also has to be modular in certain other weights (we can give a short list of possibilities). This goes back to an observation of Buzzard for $GL_2$. Previously we formulated a Serre-type conjecture on the possible weights of $\rho$. Under the assumption that the weights of $\rho$ are contained in the predicted weight set, we apply the above weight cycling argument to show that $\rho$ is modular in precisely all the nine predicted weights. This is joint work with Matthew Emerton and Toby Gee.

##### Subtle invariants and Traverso's conjectures for p-divisible groups

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 $D$. Then we will present our results on them. For instance, two deep conjectures of Traverso from mid 1970's pertain to the smallest truncations $D[p^m]$ of $D$ which determine either the Newton polygon of $D$ or even the isomorphism class of $D$. We report on two joint works. The first is with Ofer Gabber and the second with Marc-Hubert Nicole and Eike Lau. Based on results of the first work, the second proves refined versions of these two conjectures. If time permits, applications of our results and their proofs will be as well mentioned.

##### Faltings' height of CM cycles and Derivative of $L$-functions

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 suggests a direct relation between the Faltings' height of these CM cycles with the central derivative of some Rankin-Selberg $L$-function. As an application, we also give an `analytic proof' of the Gross-Zagier formula without computing the local intersection numbers at finite primes. This is a joint work with Jan Bruinier.

##### Comparison isomorphisms for $p$-adic formal schemes and applications

##### Mock modular forms

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 functions associated to indefinite quadratic forms.

In this talk, we will describe the nature of the modularity of the original mock theta functions, formulate a general definition of mock modular forms, and describe further examples. We will also consider a generalization to higher depth mock modular forms

##### Langlands functoriality and the inverse problem in Galois theory

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 representations attached to self-dual automrophic representations of $GL(n)$.

##### The subconvexity problem for ${GL}_2$

The subconvexity problem consist 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 automorphic forms (like tha arithmetic quantum unique ergodicity conjecture or the study of representations of integers by ternary quadratic forms). In this talk we will describe the complete resolution (ie. uniformly wrt. all parameters) of this problem for $GL_1$ and $GL_2$ automorphic L-functions over a general number field. The main ingredients of the proof are:

##### On the Andre-Oort conjecture

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 Zariki-closure of L is a subvariety of Hodge type of S. I will indicate a proof of this conjecture under GRH (this is joint work with A. Yafaev, relying on some work by Ullmo-Yafaev).

##### Potential automorphy for certain Galois representations to GL(n)

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 main innovation is that the result applies to Galois representations to $GL_n$, where the previous work dealt with representations to $Sp_n$; I can also dispense with certain congruence conditions which existed in the earlier work, and work over a CM, rather than a totally-real, field. The main technique is the consideration of the cohomology the Dwork hypersurface, and in particular, of pieces of this cohomology other than the invariants under the natural group action.

##### Bounding sup-norms of cusp forms

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 and the Laplacian eigenvalue. Similar techniques work for functions on other spaces, e.g. quotients of quaternion algebras. This is joint work with R. Holowinsky.

##### A "Relative" Langlands Program and Periods of Automorhic Forms

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 reductive group G (both locally and globally), where H is a spherical subgroup of G. We prove several results in this direction. Locally, the spectrum of H\G is described with the help of the dual group associated to any spherical variety by Gaitsgory and Nadler. Globally, period integrals are conjectured to be Euler products of explicit local functionals, which we compute at unramified places and show that they are equal to quotients of L-values. If time permits, I will also discuss an approach which shows that different integral techniques for representing L-functions (e.g.

##### The Rudnick-Sarnak Conjectures

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

##### p-adically completed cohomology and the p-adic Langlands program

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 representation will correspond to such an object. (Recall that only those Galois representations that are motivic, i.e. that come from geometry, are expected to correspond to classical automorphic eigenforms).

##### CM liftings of abelian varieties

##### A rigid irregular connection on the projective line

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, for any simple algebraic group. These l-adic local systems and their characteristic 0 counterparts have been constructed in some cases by Deligne and Katz. We will explain how to construct such a local system in the characteristic 0 case, uniformly for an arbitrary simple algebraic group, using the formalism of opers introduced by Beilinson and Drinfeld. Among other things, it provides an example of the geometric Langlands correspondence with wild ramification. This is joint work with Dick Gross.