Seminars and Colloquia by Series

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 David – Concordia 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 Tayou – Harvard 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 Sardari – Penn 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.

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