Seminars and Colloquia by Series

Local canonical heights and tropical theta functions

Series
Number Theory
Time
Wednesday, March 27, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Farbod ShokriehUniversity of Washington, Seattle

I will describe some connections between arithmetic geometry of abelian varieties, non-archimedean/tropical geometry, and combinatorics. For example, we give formulas for (non-archimedean) canonical local heights in terms of tropical invariants. Our formula extends a classical computation of local height functions due to Tate (involving Bernoulli polynomials).
Based on ongoing work with Robin de Jong.

Applications of Dynamical Belyi Maps

Series
Number Theory
Time
Wednesday, February 14, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bella TobinAgnes Scott College

Unicritical polynomials, typically written in the form $z^d+c$, have been widely studied in arithmetic and complex dynamics and are characterized by their one finite critical point. The behavior of a map's critical points under iteration often determines the dynamics of the entire map. Rational maps where the critical points have a finite forward orbit are called post-critically finite (PCF), and these are of great interest in arithmetic dynamics. They are viewed as a dynamical analogue of abelian varieties with complex multiplication and often display interesting dynamical behavior. The family of (single-cycle normalized) dynamical Belyi polynomials have two fixed critical points, so they are PCF by construction, and these maps provide a new testing ground for conjectures and ideas related to post-critically finite polynomials. Using this family, we can begin to explore properties of polynomial maps with two critical points. In this talk we will discuss applications of this family in arithmetic dynamics; in particular, how this family can be used to determine more general reduction properties of PCF polynomials. 

Sums of odd-ly many fractions and the distribution of primes

Series
Number Theory
Time
Wednesday, December 13, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vivian KuperbergETH

Abstract: In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: For an integer $k \ge 2$, consider the set of $k$-tuples of reduced fractions $\frac{a_1}{q_1}, \dots, \frac{a_k}{q_k} \in I$, where $I$ is an interval around $0$.
How many $k$-tuples are there with $\sum_i \frac{a_i}{q_i} \in \mathbb Z$?

When $k$ is even, the answer is well-known: the main contribution to the number of solutions comes from ``diagonal'' terms, where the fractions $\frac{a_i}{q_i}$ cancel in pairs. When $k$ is odd, the answer is much more mysterious! In ongoing work with Bloom, we prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.

Asymmetric Distribution of Extreme Values of Cubic L-functions on the 1-line

Series
Number Theory
Time
Wednesday, December 6, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chantal DavidConcordia University

A fundamental problem in analytic number theory is to calculate the maximal value of L-functions at a given point. For L-functions associated to quadratic Dirichlet characters at s = 1, the upper bounds and Ω-results of Littlewood differ by a factor of 2, and it is a long-standing (and still unsolved) problem to find out which one is closer to the truth. The important work of Granville and Soundararajan, who model the distribution of L(1, χ) by the distribution of random Euler products L(1, X) for random variables X(p) attached to each prime, shed more light to the question. We use similar techniques to study the distribution of L(1, χ) for cubic Dirichlet characters. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values. This is a joint work with P. Darbar, M. Lalin and A. Lumley.

Vanishing of Brauer classes on K3 surfaces under reduction

Series
Number Theory
Time
Wednesday, November 1, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Salim TayouHarvard University

Given a Brauer class on a K3 surface over a number field, we prove that there exists infinitely many primes where the reduction of the Brauer class vanishes, under some mild assumptions. This answers a question of Frei--Hassett--Várilly-Alvarado. The proof uses Arakelov intersection theory on GSpin Shimura varieties. If time permits, I will explain some applications to rationality questions. The results in this talk are joint work with Davesh Maulik.

Magic functions for the Smyth-Siegel trace problem

Series
Number Theory
Time
Wednesday, September 20, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Naser SardariPenn State

We study the Schur-Siegel-Smyth trace problem. We introduce a new linear programming problem that inclues Smyths' constraints, and we give an exact solution to it. This improves the best known lower bound on the Siegel trace problem which is based on Smyths' method. In a special case, we recover Siegel's original upper bound.  Our method unifies Siegel's and Smyth's work under the same framework. This is joint work with Bryce Orloski.