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

##### Curves with many symmetries

By a celebrated theorem of Hurwitz, a curve $X/{\bf C}$ 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 ${\rm X}(N)$, often have a rich structure with diverse connections to and near number theory. We review the relevant basic theorems and give some explicit examples, including recent observations on continuous families of such curves, for which the number of automorphisms is bounded by $12(g-1)$.

##### On real zeros of holomorphic Hecke cusp forms and sieving short intervals

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 introduce the problem and outline their argument that many such zeros exist if many short intervals contain numbers whose all prime factors belong to a certain subset of primes. Then I'll speak about new results on this sieving problem which lead to improved lower bounds for the number of real zeros.

##### Hodge correlators, Hodge symmetries, and Rankin-Selberg integrals

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 $\zeta$-elements, organized by Kato into an Euler system.

We define Hodge correlators - a collection of complex numbers given by certain integrals associated with a complex variety $X$. When $X$ is a modular curve, the simplest of them literally coincide with Rankin-Selberg integrals, and the entire collection consists of periods of mixed motives which are iterated extensions of the motivic $\zeta$-elements.

##### A new formulation of the Gross-Zagier formula

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.

##### 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 one is with Ofer Gabber and the second one is with Marc-Hubert Nicole and Eike Lau. Based on results of the first work, the second one proves refined versions of these two conjectures. If time permits, applications of our results and their proofs will be as well mentioned.

##### Arithmetic inner product formula

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 automorphic representations with certain epsilon factor 1. The arithmetic one concerns central L-derivatives of those with the epsilon factor -1, whose central L-values vanish automatically. This formula, which is still a conjecture for higher rank, relates the canonical height of special cycles on certain Shimura varieties and the central L-derivatives.

##### A second main term for counting cubic fields, and biases in arithmetic progressions

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 discover a curious bias in the secondary term. Roberts' conjecture has also been proved independently by Bhargava, Shankar, and Tsimerman. In contrast to their work, our proof uses the analytic theory of Sato-Shintani's zeta functions. This is a joint work with Frank Thorne. We give a generalization for counting relative cubic extensions of a given base number field, which is a joint project with Frank Thorne and Manjul Bhargava.

##### Weights in a Serre-type conjecture for U(3)

We consider a generalisation 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 Serre weights of rho. If rho is modular and irreducible locally at p we establish this conjecture, modulo weights that are close to the boundary. Under our assumptions there are 9 predicted weights. This is joint work with Matthew Emerton and Toby Gee.

##### Spectral factors in endoscopic transfer

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 generates useful test functions on the real points of an endoscopic group. I will discuss some of this, mainly to look at structure of the weights, or transfer factors, attached to pairs of conjugacy classes. Then I will introduce a parallel definition of factors based on representations instead of conjugacy classes. As in the geometric case the factors simplify when a Whittaker normalization is available. Results so far indicate that these spectral factors provide exactly the refined form of dual transfer needed globally.

##### Heights, discriminants and conductors

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 and we discuss Diophantine applications.

##### On how the first term of an arithmetic progression can influence the distribution of an arithmetic sequence

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 sequence), then the progressions a mod q will tend to contain fewer elements of the arithmetic sequence than other progressions a mod q, on average over q. The observed phenomenon is for quite small arithmetic progressions, and the maximal size of the progressions is fixed by the nature of the sequence.

##### Sup-norms, Whittaker Periods and Hypergeometric Sums

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 |f(g)| for all g in G. Then we will explain an approach to this problem via Whittaker periods. We establish a new formula for non-archimedean Whittaker functions. The formula involves 2F1 hypergeometric sums and generalizes classical results of Casselman and others. As an application we disprove a folklore conjecture on the sup-norm of GL(2) modular forms.

##### Arithmetic Fake Compact Hermitian Symmetric Spaces

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 Mumford in 1978 using p-adic uniformization. This construction is so indirect that it is hard to obtain geometric properties of the fake projective plane. A major problem in the theory of complex algebraic surfaces was to find all fake projective planes in a way which allows us to discover their geometric properties. In a joint work with Sai-Kee Yeung, which uses considerable amount of the theory of arithmetic groups, number theory and the Bruhat-Tits theory, this has been achieved.

##### Modularity Lifting in Non-Regular Weight

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 representations and automorphic representations in question have regular weight. I will describe a method to overcome this restriction in certain cases. I will focus mainly on the case of weight 1 elliptic modular forms. This is joint work with Frank Calegari.

##### The Tamagawa Number Formula Via Chiral Homology (joint with J. Lurie)

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 Atiyah-Bott formula for the cohomology of the moduli space $Bun_G(X)$ of principal G-bundles on $X$. The latter formula makes sense over an arbitrary ground field and says that $H^*(BunG(X))$ is given by the chiral homology of the commutative chiral algebra corresponding to $H^*(BG)$, where BG is the classifying space of $G$. When the ground field is $C$, the Atiyah-Bott formula can be easily proved by considerations from differential geometry, when we think of G-bundles as connections on the trivial bundle modulo gauge transformations.

##### Mod $p$ Points on Shimura Varieties

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 report on progress towards the conjecture in the case of Shimura varieties attached to non-exceptional groups.

##### Hypergeometric Motives

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 which is numerically accessible. As an illustration we will consider the case of Artin L-functions. In the main example the corresponding Galois group is a subgroup of the Weyl group of F_4 of order 1152. The associated degree four L-functions are related to the lines in certain affine cubic surfaces. This is joint work with H. Cohen.

##### Constructible Functions, Integrability, and Harmonic Analysis on p-adic Groups

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 functions". These functions specialize to fairly general functions on p-adic fields; the theory of motivic integration for such functions specializes to the classical p-adic integration when $p$ is sufficiently large. I will talk about the recent joint work with R. Cluckers and I. Halupczok, where we explore the delicate properties of constructible functions, such as integrability and boundedness. This leads to the "transfer of integrability principle" (which, in a sense, can be thought of as a long-reaching extension of the Ax-Kochen-Ersov principle).

##### Local Models of Shimura Varieties

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.

##### Local models of Shimura varieties

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.