NTS ABSTRACT

Return to NTS Spring 2017

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 3-isogeny. We also prove the analogous result for the sextic twists of j-invariant 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 Ono--Skinner (resp. Perelli--Pomykala). We prove these results by establishing a congruence formula between p-adic 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 G-extensions of F_q(t) of bounded discriminant.

Feb 2

Arul Shankar

Bounds on the 2-torsion 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 Brauer--Siegel 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 p-torsion 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" Brauer--Siegel bound. In this talk, we will discuss a proof of a subconvex bound on the size of the 2-torsion 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

L-function 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 L-function 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 Andre-Oort conjecture. Some non-abelian cases were proved by myself and others. In all proved cases, the L-function is either Dirichlet characters or quadratic Hecke characters. A natural question is what kinds of Artin L-functions 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 L-functions 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 Bhargava-Ekedahl 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 k-fold 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 30

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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 Mordell-Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K. Under suitable local constraints and finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups.

Apr 13

Eric Mortenson

Kronecker-type q-series 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 higher-dimensional Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall 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 Kronecker-type 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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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 Gross-Prasad 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 case-by-case 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.