Monday 9th January 2023: University of Oxford

Titles and abstracts:

Ben Green (Oxford)

Title: Quadratic forms in 8 prime variables

Abstract: Solving equations in prime numbers is an old topic. For instance, the ternary Goldbach problem asks one to solve the linear equation p_1 + p_2 + p_3 = N in three prime variables. 

I will discuss a recent paper of mine, the aim of which is to find prime solutions of quadratic equations in as few variables as I can, which turns out to be 8 variables.

The traditional approach to problems of this type, the Hardy-Littlewood circle method, does not quite suffice. The main new idea is to involve the Weil representation of the symplectic groups Sp_8(Z/qZ). I will explain what this is, and what it has to do with the original problem.

Rosa Winter (KCL)

Title: Weak weak approximation for del Pezzo surfaces of degree 2

Abstract: Let X be an algebraic variety over a number field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. Questions one might ask are, is X(k) empty or not? And if it is not empty, how `large' is X(k)? 

Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d≥3, these are the smooth surfaces of degree d in Pd). The lower the degree, the more complex del Pezzo surfaces are. 

After giving an overview of different notions of 'many/ rational points and what is known so far for del Pezzo surfaces, I will focus on work in progress joint with Julian Demeio and Sam Streeter on so-called weak weak approximation for del Pezzo surfaces of degree 2.

Julian Lyczak (Bath)

Title: Paucity of rational points on fibrations with multiple fibres

Abstract: I will report on an ongoing project with Tim Browning and Arne Smeets about the density of everywhere locally soluble members within a family of varieties. In this talk I will focus on the most general case, where components of higher multiplicity are allowed.

Using logarithmic geometry, I will derive exact conditions for a fibre to be locally soluble. Then using the large sieve we obtain upper bounds for the relevant counting problems for several families of examples. In some cases we can also provide lower bounds of the same order, and even asymptotics.

Using these results I will discuss different ways in which the Loughran-Smeets conjecture depends on the absence of multiple fibres.

Hanneke Wiersema (Cambridge)

Title: Modularity in the partial weight one case

Abstract: The strong form of Serre's conjecture states that a two-dimensional mod p representation of the absolute Galois group of Q arises from a modular form of a specific weight, level and character. Serre restricted to modular forms of weight at least 2, but Edixhoven later refined this conjecture to include weight one modular forms. 

In this talk we explore analogues of Edixhoven's refinement for Galois representations of totally real fields, extending recent work of Diamond–Sasaki. In particular, we show how modularity of partial weight one Hilbert modular forms can be related to modularity of Hilbert modular forms with regular weights, and vice versa. Time permitting, we will also discuss a p-adic Hodge theoretic version of this.