Junior Number Theory 2024-2025

Week 1 - 6th January 2025

Alexandros Groutides (University of Warwick) - Integrality of GL(2) x GL(2) Rankin-Selberg zeta integrals

Local zeta integrals were firstly introduced in Tate's thesis as integral representations of Euler factors associated to GL(1) L-functions of Dirichlet characters. Since then, the theory has been developed for irreducible, admissible, generic representations of various reductive groups. In particular, for GL(n) x GL(m) by Jacquet, Langlands, Shalika and Piatetskii-Shapiro, with the theory differing substantially whenever n and m coincide. In this talk we will specialize to the GL(2) x GL(2) case, or more classically, the convolution of two cusp forms. We will start from scratch, introducing the necessary tools to construct these complex integrals, and state a celebrated theorem of Jacquet-Langlands concerning absolute convergence and meromorphic continuation. Finally, we will introduce a new notion of integral test data, inspired by the work of Loeffler and Loeffler-Skinner-Zerbes on unramified zeta integrals, and see how it fits into this picture.

Week 2 - 13th January 2025

Catinca Mujdei (LSGNT) - Low-lying zeros of families of L-functions

The Katz-Sarnak philosophy aims to describe the distribution of the zeros of a family $\mathcal{F}$ of $L$-functions near the central point $s=\frac{1}{2}$, when the $L$-functions are ordered by conductor. It is conjectured that these distributions are governed by a symmetry group $G(\mathcal{F})$ related to random matrix theory. I will discuss some results that provide encouraging evidence towards these conjectures for certain families of Dirichlet $L$-functions.

Week 3 - 20th January 2025

Santiago Vázquez Sáez (LSGNT) - The Duffin-Schaeffer Conjecture

The Duffin-Schaeffer conjecture is a central result in Diophantine approximations, concerning the proportion of real numbers that can be approximated in infinitely many ways by irreducible fractions, given a specific error threshold. In this talk, we will explore the ideas behind the proof of the (quantitative) Duffin-Schaeffer conjecture, focusing on the recent argument of Hauke, Walker, and Vazquez. We will also touch on the recent improvements of Koukoulopoulos, Maynard, and Yang, who established an almost sharp bound for the number of such approximations.

Week 4 - 27th January 2025

Frederick Thøgersen (University of Nottingham) - Applications of Branching Laws on $p$-adic $L$-functions

Branching laws, the studies of how $\mathbf{G}$-representations decompose as $\mathbf{H}$-representations when restricted to a subgroup $\mathbf{H}\subset\mathbf{G}$, play a critical role in several $p$-adic $L$-function constructions. We will explore how the branching laws give us the tools to make sure the special values of an $L$-function $L(\pi,s)$ align with the twisting action on a certain representation $V$ related to an automorphic representation $\pi$. From this, we will then consider how one might use the aforementioned alignment to find the same alignment in locally analytic functions $\mathcal{A}(X,R)$ and get the desired maps in the locally analytic distributions $\mathcal{D}(X,R)=\text{Hom}_{\text{cts}}(\mathcal{A}(X,R),R)$. If time allows, I will present a result for certain $\pi$ of forms of $\text{Res}_{K/\mathbb{Q}}(\mathbf{GL}_{2n})$.

Week 5 - 3rd February 2025

Kenji Terao (University of Warwick) - Modular curves and isolated points

Modular curves are objects of central importance in arithmetic geometry, parametrizing elliptic curves with particular Galois representations. They form a key part of the proof of results such as Fermat's Last Theorem and Mazur's torsion theorem. On the other hand, isolated points are "exceptional" low-degree points on curves, which lie outside the infinite families of low-degree points effected by the geometry of the curve. In this talk, I will aim to give a gentle introduction to these two concepts, the former via a stroll through the world of moduli spaces, the latter via an examination of Faltings's proof of the Mordell conjecture. In particular, little to no prior knowledge will be assumed. Time permitting, I will conclude with some recent advances on the intersection of these two notions.

Week 6 - 10th February 2025

Sophie Maclean (Kings College London) - Pair correlations and the metric Poissonian property

This talk studies the distribution of dilated integer sequences modulo 1 i.e. the distribution of $\alpha\mathcal{A}$ mod $1$, where $\alpha\in[0,1)$ and $\mathcal{A}\subseteq\mathbb{N}$. Specifically, we look at when such sequences have a pseudorandom property. To do this, we will consider the equidistribution and the gap distributions of such sequences, looking at what has been proven, and what still remains a conjecture.

Week 7 - 17th February 2025

Khalid Younis (University of Warwick) - Smooth numbers and digit constraints

A popular topic in number theory is to investigate sequences of arithmetic interest after imposing some constraint on their digits. For example, one can arbitrarily prescribe a small fraction of digits, which simultaneously generalises the problem of counting elements in short intervals and in arithmetic progressions. We will discuss the roles of harmonic analysis and zeros of L-functions, with a focus on recent work on smooth numbers, that is, numbers whose prime factors are all small.

Week 8 - 24th February 2025

Harry Spencer (LSGNT) - Curves, representation theory and conductors

We discuss how to use tools from Galois theory and representation theory to study the arithmetic of curves and their Jacobians. In particular, we demonstrate an application to finding wild conductor exponents.

Week 9 - 3rd March 2025

Philip Holdridge (University of Warwick) - Partition Regularity and Piatetski-Shapiro

A common problem in Additive combinatorics is to show that a Diophantine equation is partition regular, that is, that for every partition of the natural numbers into finitely many sets, there is one such set in which the equation has a solution. One may also look at partitions of certain subsets of the natural numbers, such as the primes. In this talk, I will discuss the problem of showing partition regularity of linear equations in the set of Piatetski-Shapiro numbers, which are numbers of the form floor(n^c) for a fixed parameter c. We will also give an introduction to the transference principle, a method which uses Fourier analysis to "transfer" solutions from a set of positive density in the integers to solutions in a sparse set of integers. This is based on joint work with Sam Chow and Jon Chapman.

Week 10 - 10th March 2025

Alvaro Gonzalez Hernandez (University of Warwick) - Intersections of the automorphism and p-rank strata in the moduli space of genus two curves

There are two key invariants of a curve over a field of positive characteristic: its automorphism group and its p-rank. In this talk, I will define these two invariants and discuss how they are related in the case of curves of genus two. In order to do this, I will explain how to construct the moduli space of genus two curves, and how to compute both the strata of curves with a fixed automorphism and the strata of curves with a fixed p-rank. Finally, I will briefly discuss how learning about the intersections of these strata can help us understand some interesting examples of K3 surfaces in positive characteristic.