NTS ABSTRACT
Return to NTS Spring 2017
Contents
Sept 21
Chao Li 
Goldfeld's conjecture and congruences between Heegner points 
Given an elliptic curve E over Q, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever E has a rational 3isogeny. We also prove the analogous result for the sextic twists of jinvariant 0 curves. For a more general elliptic curve E, we show that the number of quadratic twists of E up to twisting discriminant X of analytic rank 0 (resp. 1) is >> X/log^{5/6}X, improving the current best general bound towards Goldfeld's conjecture due to OnoSkinner (resp. PerelliPomykala). We prove these results by establishing a congruence formula between padic logarithms of Heegner points based on Coleman's integration. This is joint work with Daniel Kriz. 
Jan 26
Jordan Ellenberg 
Upper bounds for Malle's conjecture over function fields 
I will talk about this paper
https://arxiv.org/abs/1701.04541 joint with Craig Westerland and TriThang Tran, which proves an upper bound, originally conjectured by Malle, for the number of Gextensions of F_q(t) of bounded discriminant.

Feb 2
Arul Shankar 
Bounds on the 2torsion in the class groups of number fields 
(Joint with M. Bhargava, T. Taniguchi, F. Thorne, J. Tsimerman, Y. Zhao)
Given a number field K of fixed degree n over Q, a classical theorem of BrauerSiegel asserts that the size of the class group of K is bounded by O_\epsilon(Disc(K)^(1/2+\epsilon). For any prime p, it is conjectured that the ptorsion subgroup of the class group of K is bounded by O_\epsilon(Disc(K)^\epsilon. Only the case n=p=2 of this conjecture in known. In fact, for most pairs (n,p), the best known bounds come from the "convex" BrauerSiegel bound. In this talk, we will discuss a proof of a subconvex bound on the size of the 2torsion in the class groups of number fields, for all degrees n. We will also discuss an application of this result towards improved bounds on the number of integral points on elliptic curves. 
Feb 9
Tonghai Yang 
Lfunction aspect of the Colmez Conjecture 
Associate to a CM type, Colmez defined two invariants: Faltings height of the associated CM abelian varieties of this CM type, and the log derivative of some mysterious Artin Lfunction nifonstructed from this CM type. Furthermore, he conjectured them to be equal and proved the conjecture for Abelian CM number fields (up to log 2). The average version of the conjecture was proved recently by two groups of people which has significant implication to AndreOort conjecture. Some nonabelian cases were proved by myself and others. In all proved cases, the Lfunction is either Dirichlet characters or quadratic Hecke characters. A natural question is what kinds of Artin Lfunctions show up in this conjecture. In this talk, we will talk about some interesting examples in this. This is a joint work with Hongbo Yin. 
Feb 16
Alexandra Florea 
Moments of Lfunctions over function fields 
I will talk about the moments of the family of quadratic Dirichlet L–functions over function fields. Fixing the finite field and letting the genus of the family go to infinity, I will explain how to obtain asymptotic formulas for the first four moments in the hyperelliptic ensemble.

Feb 23
Dongxi Ye 
Borcherds Products on Unitary Group U(2,1) 
In this talk, I will first briefly go over the concepts of Borcherds products on orthogonal groups and unitary groups. And then I will present a family of new explicit examples of Borcherds products on unitary group U(2,1), which arise from a canonical basis for the space of weakly holomorphic modular forms of weight $1$ for $\Gamma_{0}(4)$. This talk is based on joint work with Professor Tonghai Yang.

Mar 2
Frank Thorne 
Levels of distribution for prehomogeneous vector spaces 
One important technical ingredient in many arithmetic statistics papers is
upper bounds for finite exponential sums which arise as Fourier transforms of characteristic functions of orbits. This is typical in results obtaining power saving error terms, treating "local conditions", and/or applying any sort of sieve. In my talk I will explain what these exponential sums are, how they arise, and what their relevance is. I will outline a new method for explicitly and easily evaluating them, and describe some pleasant surprises in our end results. I will also outline a new sieve method for efficiently exploiting these results, involving Poisson summation and the BhargavaEkedahl geometric sieve. For example, we have proved that there are "many" quartic field discriminants with at most eight prime factors. This is joint work with Takashi Taniguchi.

Mar 9
Brad Rodgers 
Sums in short intervals and decompositions of arithmetic functions 
In this talk we will discuss some old and new conjectures about the behavior of sums of arithmetic functions in short intervals, along with analogues of these conjectures in a function field setting that have been proved in recent years. We will pay particular attention to some surprising phenomena that comes into play for the kfold divisor function, and a decomposition of arithmetic functions in a function field setting that helps elucidate what's happening. I also hope to discuss the relation of these results to symmetric function theory and a connection to algebraic geometry in a recent paper of Hast and Matei.

Mar 16
Mar 30
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abstract

Apr 6
Celine Maistret 
Parity of ranks of abelian surfaces 
Let K be a number field and A/K an abelian surface (dimension 2 analogue of an elliptic curve). By the MordellWeil theorem, the group of Krational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and SwinnertonDyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the Lseries determines the parity of the rank of A/K. Under suitable local constraints and finiteness of the ShafarevichTate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups. 
Apr 13
Eric Mortenson 
Kroneckertype qseries identities and formulas for sums of squares and sums of triangular numbers 
We recall Kronecker's identity and review how limiting cases give the representations of a number as a sum of four squares and the representations of a number as a sum of two squares. The two formulas imply respectively Lagrange's theorem that every number can be written as a sum of four squares and Fermat's theorem that an odd prime can be written as the sum of two squares if and only if it is congruent to 1 modulo 4. By considering a limiting case of a higherdimensional Kroneckertype identity, we obtain an identity found by both Andrews and Crandall. We then use the AndrewsCrandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares. From the Kroneckertype identity, we also deduce Gauss's theorem that every positive integer is representable as a sum of three triangular numbers. 
Apr 20
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abstract

Apr 27
Yueke Hu 
Mass equidistribution of cusp forms on torus in depth aspect 
In this talk I will talk about mass equidistribution of cusp forms of level $p^{c}$ when restricted to geodesics or Heegner points as $c$ goes to infinity. A key ingredient is a discussion of the test vector for Waldspurger’s period integral, generalizing the GrossPrasad test vector.

May 4
Yiannis Sakellaridis 
Stacks, regularization of orbital integrals, and the relative trace formula 
The relative trace formula of Jacquet is a putative generalization of the Arthur–Selberg trace formula, which is being used to establish functoriality and relations between periods of automorphic forms, as the trace formula is being used to establish functoriality and character relations. As of now, it has been developed only on a casebycase basis, with methods that are similar but, to some extent, ad hoc. I will describe a general approach to the geometric side of the relative trace formula, which in many cases provides the correct answer. The approach has a local and a local component: Locally, one develops a notion of "Schwartz space of a quotient stack", the space of "test functions" for the relative trace formula where pure inner forms of the group naturally show up. Globally, and quite independently, one develops a theory of regularization of orbital integrals that is based on toric geometry. I will also explain why this purely geometric approach is not enough to produce an answer in some cases (such as the original Arthur–Selberg trace formula), and will give some hints on what might be done in those cases.
