RESEARCH TALKS

Shabnam Akhtari (Pennsylvania State University)

A Quantitative Primitive Element Theorem
 
Abstract
Let K be an algebraic number field. The Primitive Element Theorem implies that the number field K can be generated over the field of rational numbers by a single element of K. We call such an element a generator of K. A simple and natural question is “what is the smallest generator of a given number field?” (and how to find it!) In order to express this question more precisely, we will introduce some height functions. Then we will discuss some open problems and some recent progress in this area.

Lior Bary-Soroker (Tel-Aviv University)

Hilbert’s irreducibility theorem over algebraic groups
 
Abstract

Hilbert’s irreducibility theorem is a central theorem in arithmetic geometry with numerous applications. A simple quantitative version of the theorem says that if f(x,y) is an irreducible polynomial in 2 variables with rational coefficients, then for almost all integers a, the univariate polynomial f(a,y) is irreducible. In recent years there has been a surge of studies on variants of Hilbert’s irreducibility theorem over algebraic groups; that is, replacing x by the coordinates of an algebraic group G. The talk aims to explore these variants, highlighting their connections to analytic number theory.

Sandro Bettin (Università degli studi di Genova)

The distribution of modular symbol and of Fourier series with modular coefficients
 
Abstract

Using methods from dynamical systems, we determine the asymptotic distribution of modular symbols associated to modular forms of arbitrary weight and level. This includes in particular the case of Eisentein series of arbitrary level which was missing from earlier works on modular symbols of Petridis and Risager, Nordentoft, Lee and Sun, and Bettin and Drappeau. As an application we also determine the distribution of Fourier series with modular coefficients. This is joint work with Sary Drappeau and Jungwon Lee.

Alina Bucur (University of California, San Diego)

Counting points on curves over finite fields