Junior Number Theory Seminar

Hello,

Welcome to Warwick's Junior Number Theory Seminar!

The seminar takes place during term time on Mondays from 11am-12pm in

B3.01 (terms 1 and 2),
B3.02 (term 3).

The seminar is organised by Joseph Harrison and Paweł Nosal.

If you want to give a talk in this seminar, contact one of the organisers!

Term 1

Date	Speaker	Title
6th October 2025	Fred Tyrrell (Bristol) (Cancelled)	Bounded exponential sums with multiplicative coefficients
13th October 2025	Ruth Raistrick (Glasgow)	Galois module structure of unit groups of rank 2 over rank 1
20th October 2025	Cédric Pilatte (Oxford)	Two-point Liouville correlations using non-backtracking matrices
27th October 2025	Alexandros Groutides (Warwick)	From Dirichlet series to automorphic $L$-functions
3rd November 2025	Ashleigh Ratcliffe (Leicester)	A systematic approach to solving Diophantine equations
10th November 2025	Alberto Acosta Reche (UCL)	Chebotarev geodesic theorem
17th November 2025	Kenji Terao (Warwick)	Computing class groups of global fields
24th November 2025	Maryam Nowroozi	Del Pezzo surfaces with one bad prime over cyclotomic $\mathbb{Z}_{\ell}$-extensions
1st December 2025	Ayesha Bennett (Cambridge)	TBD
8th December 2025	Fred Tyrrell (Bristol)	TBD

Abstracts

Week 1 - 6th October 2025

Fred Tyrrell (University of Bristol) - Bounded exponential sums with multiplicative coefficients

(Cancelled)

In this talk, we explore when the exponential sum $\sum_{n \leq x} f(n)e(n\alpha)$ can be bounded, where $f$ is a multiplicative function and $\alpha$ is some real number. After motivating the problem with classical number theory and recent developments in discrepancy theory, I will then state our main classification results, and give some explanation. If there is time, I will also say a few words about the ideas involved, and possible future directions of research. This is joint work with Pierre-Alexandre Bazin and Ihor Pylaiev.

Week 2 - 13th October 2025

Ruth Raistrick (University of Glasgow) - Galois module structure of unit groups of rank 2 over rank 1

Arithmetic statistics is the study of "the proportion" of arithmetic objects. That is to say, we count certain objects up to some bound, in this case we count a certain Galois module structure of the free part of the units of a number field. Say we have $K$ a number field of unit rank 1 and $L$, an extension of $K$ of unit rank 2, the free part of the unit group has the natural structure of a $\text{Gal}(L/K)$-module, given we only have two possible such structures, we ask how often we see one over the other.

Week 3 - 20th October 2025

Cédric Pilatte (University of Oxford) - Two-point Liouville correlations using non-backtracking matrices

The two-point Chowla conjecture predicts that $\sum_{n \leq x} \lambda(n)\lambda(n+1) = o(x)$ as $x\to \infty$, where $\lambda : \mathbb{N} \to \{\pm 1\}$ is the Liouville function (the completely multiplicative function defined by $\lambda(p)=-1$ at every prime $p$). While this remains an open problem, weaker versions of this conjecture have been proved in the last decade. In 2021, Helfgott and Radziwiłł obtained the estimate

$$\sum_{n \leq x} \frac{1}{n} \lambda(n)\lambda(n+1) \ll \frac{\log x}{(\log \log x)^{1/2}}$$

for logarithmically weighted correlations, by studying the eigenvalues of a symmetric matrix encoding divisibility by small primes.

In this talk, I will present a new approach, based on the spectral analysis of certain non-symmetric matrices called ``non-backtracking matrices''. Using this strategy, we obtain the improved bound

$$\sum_{n \leq x} \frac{1}{n} \lambda(n)\lambda(n+1) \ll (\log x)^{1-c}$$

where $c>0$ is an absolute constant.

Week 4 - 27th October 2025

Alexandros Groutides (University of Warwick) - From Dirichlet series to automorphic $L$-functions

Given a Dirichlet character $\chi$ there is a natural candidate for its associated $L$-function $L(\chi,s)$ in terms of the familiar Dirichlet series. The same can be said for a holomorphic modular form, using the theory of $q$-expansions. Things quickly become a lot more complicated when one attempts to construct an $L$-function for the Rankin-Selberg convolution of two modular forms. This is where the approach of Jacquet-Langlands really shines. The talk will be a gentle introduction to the above.

Week 5 - 3rd November 2025

Ashleigh Ratcliffe (University of Leicester) - A systematic approach to solving Diophantine equations

This talk reports on the following project: define a suitable notion of "size'' of a polynomial Diophantine equation, order all equations by size, and solve them in this order. On the way, we discuss how to solve Diophantine equations using Vieta jumping, Jacobi symbols, the Mordell-Weil sieve, develop a general method to solve all two-variable three-monomial equations, and deduce the complete list of integers $|n| \leq 200$ which are representable as the difference of two fourth powers.

Week 6 - 10th November 2025

Alberto Acosta Reche (UCL) - Chebotarev geodesic theorem

Several mathematicians (Hejhal/Huber/Selberg/...) noticed an analogy between prime numbers and primitive closed geodesics on a finite-volume hyperbolic surface. The counting result that is the geodesic analogue of the prime number theorem is usually called the prime geodesic theorem. In his PhD thesis, Sarnak proved a geodesic analogue of the Chebotarev density theorem, which we call the Chebotarev geodesic theorem. In this talk, I will present recent ongoing work which improves the best-known error term for the Chebotarev geodesic theorem on the modular surface.

Week 7 - 17th November 2025

Kenji TeraoLink opens in a new window (University of Warwick) - Computing class groups of global fields

The class group of a number field or global function field is an important invariant throughout number theory, with connections to class field theory, arithmetic geometry, cryptography and certain reading groups, to mention a few. In this talk, we will explore some of the methods used to compute class groups in practice, and in particular, the index calculus algorithm. Throughout, we shall also reflect on the tight connection between number fields and global function fields, which underpins results such as the Riemann hypothesis for curves over finite fields.

Week 8 - 24th November 2025

Maryam Nowroozi

Let $K$ be a number field and $S$ a finite set of primes of $K$. Scholl proved that there are only finitely many $K$-isomorphism classes of del Pezzo surfaces of any degree $1 \leq d \leq 9$ over $K$ with good reduction away from $S$.

Let instead $K$ be the cyclotomic $\mathbb{Z}_5$-extension of $\mathbb{Q}$. We show for $d=3, 4$, that there are infinitely many $\mathbb{Q}$-isomorphism classes of del Pezzo surfaces, defined over $K$, with good reduction away from the unique prime above $5$.

Week 9 - 1st December 2025

Ayesha Bennett (University of Cambridge) - TBD

Week 10 - 8th December 2025

TBD

Term 1

Date

Speaker

Title

Abstracts