Monday 17th April 2023: University of Exeter

Demi Allen (Exeter)

Title: Diophantine Approximation for systems of linear forms - some comments on inhomogeneity, monotonicity, and primitivity

Abstract: Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. In the most classical setting, a ψ-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function ψ. Khintchine’s Theorem provides a beautiful characterisation of the Lebesgue measure of the set of ψ-well- approximable numbers and is one of the cornerstone results of Diophantine Approximation. In this talk I will discuss the generalisation of Khintchine’s Theorem to the setting of approximation for systems of linear forms. I will focus mainly on the topic of inhomogeneous approximation for systems of linear forms. Time permitting, I may also discuss approximation for systems of linear forms subject to certain primitivity constraints. This talk will be based on joint work with Felipe Ramírez (Wesleyan, US).  

Ross Paterson (Bristol)

Title: Higher Genus Theory

Abstract: Gauss' genus theory, more precisely his study of `ambiguous quadratic forms', describes the 2-torsion in class groups of quadratic fields.  Gauss' observations were later made formal by the ambiguous class number formula of Hilbert, which computes the size of the two torsion in these class groups in terms of local data.  Genus theory has been generalised substantially, and today `genus theory' usually refers to the work of Frohlich, who defined the notion of `genus groups' for relative extensions K/F of number fields.  Similarly, this still provides a local formula for certain pieces of class groups.  One thinks of this formula as a `predictable' part of the class group, interfering with their, a priori random, behaviour.

In this talk I will discuss work in progress on a new generalisation of genus theory.  This higher genus theory provides a notion of genus group to selmer groups of finite Galois modules.  We will discuss how this perspective provides more refined data for class groups, as well as applications in the world of abelian varieties.

Rachel Newton (KCL)

Title: Evaluating the wild Brauer group 

Abstract: The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points. The Brauer-Manin pairing cuts out a subset of the adelic points, called the Brauer-Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer-Manin set is empty then we say there's a Brauer-Manin obstruction to the existence of rational points on X. Computing the Brauer-Manin pairing involves evaluating elements of the Brauer group of X at local points. If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For elements of order a power of p, this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Kato. Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer-Manin obstruction. This is joint work with Martin Bright.

Akshat Mudgal (Oxford)

Title: Unbounded expansion of polynomials and products

Abstract: Given natural numbers d and s, a finite set A of integers and non-constant polynomials f_1, ..., f_s in Z[x] with degrees bounded above by d, we show that either the set A generates many distinct s-fold products or many distinct sums of the form f_1(a_1) + … + f_s(a_s). In particular, writing 

A^{s} = { a_1…a_s : a_1, .., a_s in A},

we have that

|A^s| + |f_1(A) + … + f_s(A)| >> |A|^c ,

where c = c(s,d) goes to infinity with s. We also prove a corresponding result for products of polynomials. These generalise and strengthen previous results of Bourgain–Chang, Pálvölgyi–Zhelezov and Hanson–Roche-Newton–Zhelezov.