Events

Félix Baril Boudreau, University of Lethbridge

It is known since the works of Ogg (1962) and Shafarevich (1961) (under tameness assumptions), followed by Grothendieck (1968), that the rank of a given abelian variety over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on reduction data. This bound is "geometric" in nature. In particular, it holds if we replace the constant field by its algebraic closure.

Ulmer asked in 2004 if, for an elliptic curve, there was an arithmetic bound that could improve on the geometric one. This question recently obtained a positive answer (Gillibert and Levin, 2022).

In this talk, we present a similar arithmetic refinement of the geometric bound for higher-dimensional abelian varieties. When specialized to elliptic curves, we improve on Gillibert-Levin's bound. Time permitting, we will discuss some consequences of our result.

This is joint work with Jean Gillibert and Aaron Levin.

MC 5501

Nathan Grieve, Acadia University

I will report on a collection of recent results and ongoing work that surround extensions and applications of Schmidt's Subspace Theorem and Vojta's height inequalities.  As two examples: (i) It is of interest to understand the qualitative features of Diophantine exceptional sets; (ii) It is of interest to understand the extent to which algebraic points of a given bounded degree in a given general type projective variety are not-Zariski dense.  As I will explain, there are several logically equivalent points of departure for these results.  They build on a collection of my past contributions in addition to work of many others.

Online talk: https://uwaterloo.zoom.us/j/98950813087?pwd=SEl1NlNqNHl0QzlYNGJzeDVla204QT09

Huixi Li, Nankai University

In 1950, Erd\H{o}s posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough in 2015. In this presentation, we will discuss the analogous minimum modulus problem for $\mathbb{F}_q[x]$. We proof that the smallest degree of the moduli in any covering system for $\mathbb{F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. This is a joint work with Shaoyun Yi, Biao Wang, and Chunlin Wang. 

Zoom: https://uwaterloo.zoom.us/j/98950813087?pwd=SEl1NlNqNHl0QzlYNGJzeDVla204QT09

Arul Shankar, University of Toronto

Ranks of elliptic curves are often studied via their 2-Selmer groups. It is known that the average size of the 2-Selmer group of elliptic curves is 3, when the family of all elliptic curves is ordered by (naive) height. On the computational side, Balakrishnan, Ho, Kaplan, Spicer, Stein, and Weigand collect and analyze data on ranks, 2-Selmer groups, and other arithmetic invariants of elliptic curves, when ordered by height. Interestingly, they find a persistently smaller average size of the 2-Selmer group in the data. Thus it is natural to ask whether there exists a second order main term in the counting function of the 2-Selmer groups of elliptic curves. In this talk, I will discuss joint work with Takashi Taniguchi, in which we prove the existence of such a secondary term.

MC 5501

Kiseok Yeon, Purdue University

In this talk, we introduce a framework via the circle method in order to confirm the Hasse principle for random homogeneous polynomials in thin sets. We first give a motivation for developing this framework by providing an overall history of the problems of confirming the Hasse principle for homogeneous polynomials. Next, we provide a sketch of the proof of our main result and show a part of the estimates used in the proof. Furthermore, by using our recent joint work with H. Lee and S. Lee, we discuss the global solubility for random homogeneous polynomials in thin sets.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Kunjakanan Nath, University of Illinois, Urbana-Champaign

One of the key problems in number theory is to understand the correlation between two arithmetic functions. In general, it is an extremely difficult question and often leads to famous open problems like the Twin Prime Conjecture, the Goldbach Conjecture, and the Chowla Conjecture, to name a few. In this talk, we will discuss a few binary correlation problems involving primes, square-free integers, and integers with restricted digits. The objective is to demonstrate the application of Fourier analysis (aka the circle method) in conjunction with the arithmetic structure of the given sequence and the bilinear form method to solve these problems.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Alex Cowen, Harvard University

What correlation is there between the number of divisors of N and the number of divisors of N+1? This is known as the classical additive divisor problem. This talk will be about a generalized form of this question: I'll give asymptotics for a shifted convolution of sum-of-divisors functions with nonzero powers and twisted by Dirichlet characters. The spectral methods of automorphic forms used to prove the main result are quite general, and I'll present a conceptual overview. One step of the proof uses a less well-known technique called "automorphic regularization" for obtaining the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable.

MC 5417

Alex Cowan, Harvard University

What correlation is there between the number of divisors of N and the number of divisors of N + 1? This is known as the classical additive divisor problem. This talk will be about a generalized form of this question: I’ll give asymptotics for a shifted convolution of sum-of-divisors functions with nonzero powers and twisted by Dirichlet characters. The spectral methods of automorphic forms used to prove the main result are quite general, and I’ll present a conceptual overview. One step of the proof uses a less well-known technique called “automorphic regularization” for obtaining the spectral decomposition of a combination of Eisenstein series which is not obviously square-integrable.

MC 5417

Andrés Chirre, Pontifical Catholic University of Peru

In this talk, we will discuss a well-known formula of Ramanujan and its relationship with the partial sums of the Möbius function. Under some conjectures, we analyze a finer structure of the involved terms. It is a joint work with Steven M. Gonek.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Peter Oberly, University of Rochester

The Arakelov--Zhang pairing (also called the dynamical height pairing) is a kind of dynamical distance between two rational maps defined over a number field. This pairing has applications in arithmetic dynamics, especially as a tool to study the preperiodic points common to two rational maps. We will discuss some bounds on the Arakelov-Zhang pairing of f and g in terms of the coefficients of f and investigate some simple consequences of this result. 

MC 5417