Wednesday 6th April 2022: University of Bath
Oleksiy Klurman (Bristol).
Title: A tale of Fekete polynomials.
Abstract: Since their discovery by Dirichlet in the nineteenth century, Fekete polynomials (with coefficients being Legendre symbols) and their zeros attracted considerable attention, in particular, due to their intimate connection with putative Siegel zero and small class number problem. The goal of this talk is to discuss what we knew, know, and would like to know about zeros of such (and related) polynomials. Joint work with Y. Lamzouri and M. Munsch.
Fabio Ferri (Exeter).
Title: Leopoldt-type theorems for non-abelian extensions of Q.
Abstract: We prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give necessary and sufficient conditions for the ring of integers to be free over its associated order in the rational group algebra Q[G].
Valeriya Kovaleva (Oxford).
Title: Correlations of Riemann Zeta on the critical line.
Abstract: In this talk we will discuss one of the ways to study the behaviour of the Riemann zeta on the critical line. In particular, we will consider the correlations of the Riemann Zeta in various ranges. We will also explain how it is related to Motohashi's formula for the fourth moment, the moments of moments of the Riemann Zeta, and its maximum in short intervals.
Tim Dokchitser (Bristol).
Title: Constructing Galois groups over Q.
Abstract: The Inverse Galois Problem asks whether every finite group G occurs as a Galois group over Q, and, stronger, over Q(t) with no constant subfields. The constructive version of this problem also asks to produce polynomials with the right Galois group. We discuss some old and new methods and new results, including realisations of some previously unknown groups, and families for all transitive groups of degree <16 over Q(t) and all groups of order <128 over Q. The algorithms discussed in the talk have been implemented in the Magma package available from https://people.maths.bris.ac.uk/~matyd/InvGal/ together with a list of open problems on the computational aspects of the Inverse Galois Problem.