January 2024: University of Bath

Elyes Boughattas (Bath)

Title: Fibration method over function fields of curves

Abstract: Determining whether a given diophantine equation has a solution is a wide open question in number theory. For some varieties -- e.g. quadrics -- the existence of local points is enough to determine the existence of global points: this is known as the Hasse principle. Nevertheless, the latter does not hold for cubic forms, as shown by Selmer in 1951. Manin introduced in 1970 a set called the Brauer-Manin set, which is expected to describe all obstructions to the Hasse principle for rationally connected varieties.

In this talk, I shall present a work in progress which explains how this Brauer-Manin setting is related to fibrations over P^1, whenever the base field is the function field of a curve over a large finite field.

Holly Green (Bristol)

Title: On the parity conjecture for elliptic curves

Abstract: I will present a new method to compute the parity of the rank of an elliptic curve and will comment on how this construction generalises to Jacobians of curves. This method involves studying the local arithmetic attached to covers of the curve. In addition, I will discuss applications to the Birch and Swinnerton-Dyer conjecture, including a new proof of the parity conjecture for elliptic curves. This is joint work with Vladimir Dokchitser, Alexandros Konstantinou, Céline Maistret and Adam Morgan.

Lasse Grimmelt (Oxford)

Title: A new automorphic black-box for analytic number theory

Abstract: The spectral theory of automorphic forms finds remarkable applications in analytic number theory. Notably, it is utilised in results concerning the distribution of primes in large arithmetic progressions and in questions on variants of the fourth moment of the zeta function. Traditionally, these problems are addressed by reducing them to sums of Kloosterman sums, followed by either the use of existing black-box results or by-hand application of spectral theory through Kuznetsov's formula.

In this presentation, based on joint work with Jori Merikoski, I will introduce an alternative approach that entirely circumvents the need for Kloosterman sums. This approach offers increased flexibility compared to existing black-box methods, without requiring more automorphic understanding. As an application, I will present novel results on correlations of the divisor function in arithmetic progressions.

Chris Lazda (Exeter)

Title: Good reduction of Kummer surfaces modulo 2

Abstract: In residue characteristic different from 2, the good reduction of a given Kummer surface is (essentially) equivalent to the good reduction of the abelian surface it is constructed from, and is therefore completely understood via the classical Neron-Ogg-Shafarevich criterion. In residue characteristic 2, however, things are a little bit more interesting, and good reduction of a given abelian surface is not enough to ensure good reduction of its associated Kummer surface. In this talk I will explain a more refined criterion for the good reduction of K3 surfaces in general, and use it to solve the problem of good reduction for Kummer surfaces associated to abelian surfaces with good, non-supersingular reduction in residue characteristic 2. This is joint work with Alexei Skorobogatov.