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

##### Modular forms and optimization in Euclidean space

In this talk we will show how modular forms can be applied to energy minimization problems in Euclidean space. Namely, we will explain Cohn-Elkies linear programming method and present explicit constructions of

corresponding certificate functions. In particular, we will discuss optimization in dimensions 8 and 24.

##### Recent progress on Serre weight conjectures

I will discuss some recent results on Serre weight conjectures in dimension >2, based on the study of certain tame type deformation rings. This is joint work with (various subset of) D. Le, B. Levin and S. Morra.

##### Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals, we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive $p$-adic group tend to zero uniformly for every noncentral semisimple element. This is a joint work with Shin and Templier.

##### The Unpolarized Shafarevich Conjecture for K3 Surfaces

Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves over K (a number field) with good reduction outside S (a fixed finite set of primes) is finite. Faltings proved this and the analogous conjecture for abelian varieties of given degree. Zarhin proved this finiteness across all degrees. Using Faltings’ theorem, Andre proved the finiteness of K3 surfaces (over K, S) of a given degree. We prove the analog of Zarhin’s theorem, i.e. there are still finitely many K3 surfaces across all degrees.

##### Local points of supersingular elliptic curves on Z_p-extensions

Work of Kobayashi and Iovita-Pollack describes how local points of supersingular elliptic curves on ramified Z_p-extensions of Q_p split into two strands of even and odd points. We will discuss a generalization of this result to Z_p-extensions that are localizations of anticyclotomic Z_p-extensions over which the elliptic curve has non-trivial CM points.

##### The Hasse-Weil zeta functions of the intersection cohomology of minimally compactified orthogonal Shimura varieties

Initiated by Langlands, the problem of computing the Hasse-Weil zeta functions of Shimura varieties in terms of automorphic L-functions has received continual study. We will discuss how recent progress in various aspects of the field has allowed the extension of the project to some Shimura varieties not treated before. In the particular case of orthogonal Shimura varieties, we discuss the computation of the Frobenius-Hecke traces on the intersection cohomology of their minimal compactifications, and the comparison to the Arthur-Selberg trace formula via the process of stabilization. Key ingredients include comparing Harish Chandra character formulas to Kostant's Theorem on Lie algebra cohomology, and a comparison between different normalizations of the transfer factors for real endoscopy to get all the signs right.

##### The Arithmetic of Noncongruence Subgroups of SL(2,Z)

After beginning by giving a brief overview of how one can think of noncongruence modular curves as moduli spaces of elliptic curves with G-structures, we will discuss how these moduli interpretations fits into the greater body of knowledge concerning noncongruence subgroups, in particular focusing on the Unbounded Denominators Conjecture for their modular forms.

##### Albanese of Picard modular surfaces, and rational points

This is a report on a work in progress in collaboration with Dinakar Ramakrishnan. A celebrated result of Mazur states that open modular curves of large enough level do not have rational points. We study analogous questions for the Picard modular surfaces, which are Shimura varieties for a unitary group in 3 variables defined over an imaginary quadratic field M. For each M we provide specific levels for which those surfaces are Mordellic, i.e. have only finitely many points over any number field, and we have a project aiming at showing the non-existance of M-rational points at deep enough level.

##### Nonabelian Cohen-Lenstra Heuristics and Function Field Theorems

The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields. Since, by class field theory, the class group is the Galois group of the maximal unramified abelian extension, we can consider the Galois group of the maximal unramified extension as a non-abelian generalization of the class group. We will explain non-abelian analogs of the Cohen-Lenstra heuristics due to Boston, Bush, and Hajir and work, some joint with Boston, proving cases of the non-abelian conjectures in the function field analog.

##### Integral points on moduli schemes and Thue equations

We will explain a way how one can use moduli schemes and their natural forgetful maps in the study of certain classical Diophantine problems (e.g. finding all integral points on hyperbolic curves). To illustrate and motivate the strategy, we consider the case of cubic Thue equations and we discuss a joint project with Matschke in which we solved many cubic Thue equations.

##### Arithmetic and Geometry of Picard modular surfaces

Of interest are (i) the conjecture of Bombieri (and Lang) that for any smooth projective surface X of general type over a number field k, the set X(k) of k-rational points is not Zariski dense, and (ii) the conjecture of Lang that X(k) is even finite if in addition X is hyperbolic, i.e., there is no non-constant holomorphic map from the complex line C into X(C). We can verify them for the Picard modular surfaces X which are smooth toroidal compactifications of congruence quotients Y of the unit ball in C^2. We will describe an ongoing program, with Mladen Dimitrov, to prove moreover that for suitable deep levels, Y has no rational points over the natural field of definition k.

##### On the spectrum of Faltings' height

The arithmetic complexity of an elliptic curve defined over a number field is naturally quantified by the (stable) Faltings height. Faltings' spectrum is the set of all possible Faltings' heights. The corresponding spectrum for the Weil height on a projective space and the Neron-Tate height of an Abelian variety is dense on a semi-infinite interval. We show that, in contrast, Faltings' height has 2 isolated minima. We also determine the essential minimum of Faltings' height up to 5 decimal places. This is a joint work with Jose Burgos-Gil and Ricardo Menares.

##### Superconnections and special cycles

I will start by explaining Quillen's notion of a superconnection, and then will use it to define some natural

differential forms on period domains parametrizing Hodge structures. For hermitian symmetric domains, we will show that this construction recovers the forms introduced by Kudla and Millson. We will discuss the properties of these forms and how they allow to generalize results on special cycles in Shimura varieties to arithmetic quotients of period domains.

##### Diophantine Problems and the p-adic Torelli Map

We explore the comparison isomorphism of p-adic Hodge theory in the case of elliptic curves, and discuss some ideas which may be used to prove the S-unit theorem and the finiteness of rational points on higher-genus curves (Faltings' theorem).

##### 16-rank of class groups of quadratic number fields

We will discuss how Vinogradov's method applies to the study of the 2-part of class groups in certain thin families of quadratic number fields. We will show how the method yields a density result for the 16-rank in the family of quadratic number fields of discriminant -8p with p a prime. We will also explain certain limitations of the method arising from reliance on short character sum estimates.

##### The subconvexity problem

**Please note unusual start time (5:00 pm.) **The importance of the subconvexity problem is well-known. In this talk, I will discuss a new approach to establish subconvex bounds for automorphic L-functions. The method is based on adopting the circle method to separate oscillatory factors involved in the L-function. This has proved to be quite effective in dealing with a broad class of subconvexity problems, including some involving degree three L-functions. I will also mention some new applications of this method.

##### Real structures on ordinary Abelian varieties

The "moduli space" for principally polarized complex n dimensional Abelian varieties with real structure (that is, anti-holomorphic involution) may be identified with a certain locally symmetric space for the group GL(n) over the real numbers. Is it possible to make sense of the points of this space over a finite field? This talk describes joint work with Yung-sheng Tai (Haverford College) in which we propose an approach to this question for ordinary Abelian varieties.

##### On small sums of roots of unity

Let k be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of k roots of unity can be. If the roots of unity have order dividing N, then an elementary argument shows that the modulus decreases at most exponentially in N (for fixed k). Moreover it is known that the decay is at worst polynomial if k<5. But no general sub-exponential bound is known if k>=5. In this talk I will present evidence that the modulus decreases at most polynomially for prime values of N by showing that counterexamples must be very sparse. We do this by counting rational points that approximate a set that is definable in an o-minimal structure. This is motivated by the counting results of Bombieri-Pila and Pila-Wilkie.

##### Mirror symmetry and another look at Kloosterman sums

I have been developing a new bridge between number theory and symplectic geometry. The special program at the IAS and a workshop this week in Wolfensohn Hall are devoted to mirror symmetry. I will describe this bridge, explain that there are travel lanes in both directions, and discuss what the benefits are for us number theorists. I will focus in this talk on the generalized Kloosterman sums studied by Deligne, Dwork, Katz, Gross and Heinloth-Ngo-Yun, and which we relate to the Dubrovin-Givental quantum connection of flag varieties. Work with Thomas Lam.

##### Galois Representations for the general symplectic group

In a recent preprint with Sug Woo Shin (https://arxiv.org/abs/1609.04223) I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.