Number Theory Seminar

The Number Theory Seminar is a 55-minute in-person research seminar, including questions. Talks are suitable for PhD students (including new PhD students) from all fields of number theory.

The seminar is at 15:00 on Mondays (except bank holidays). On seminar days we meet for lunch at 12:30 and coffee at 16:00 in the common room. Seminars are held in B3.02 of the Zeeman building. Sometimes a different room is used, see the entry below.

Colleagues and especially number theory group members are warmly encouraged to suggest speakers by emailing the organisers: Simon Rydin Myerson (term 1), Adam Harper (terms 2-3), Harry Schmidt (all terms).

We kindly remind members that it is polite to the speaker, to come to talks we might not personally expect to be interested in, and in compensation there will be a good audience for the speakers each of us is interested in!

A list of members of the group and research interests is available.

Click on a title to view the abstract!

Upcoming Seminars

Past Seminars

10 March 2025 at 15:00 in B3.02
Speaker: Oleksiy Klurman (University of Bristol)
Title: SEMINAR POSTPONED DUE TO SPEAKER ILLNESS

Abstract: .

07 March 2025 at 13:00 in B3.02
Speaker: David Masser (Basel)
Title: TBA

Abstract: TBA

24 February 2025 at 15:00 in B3.02
Speaker: Shin-ya Koyama (Toyo University)
Title: Weighted Prime Number Theorem with Refinements

Abstract: For $s \geq 0$ , the weighted counting functions for primes $p$ with weight $p^{-s}$ are defined as $ \pi_s(x) = \sum_{p \leq x} p^{-s}, \quad \pi_s(x, q, a) = \sum_{\substack{p \equiv a \ (\text{mod} \ q) \\ p \leq x}} p^{-s}. $ When $s = 0$ , they agree to the classical counting functions $\pi(x)$ and $\pi(x, q, a)$ . In 2023, Aoki and Koyama proved under the Deep Riemann Hypothesis for Dirichlet $L$ -functions that $\begin{equation*} \pi_s(x, q, b) - \pi_s(x, q, a) = \begin{cases} c_q \log \log x + O(1), & (x \to \infty, s = \frac{1}{2}) \\ O(1), & (x \to \infty, s > \frac{1}{2}) \end{cases} \end{equation*}$ for any $a \in (\mathbb{Z}/q\mathbb{Z})^{\times 2}$ , $b \in (\mathbb{Z}/q\mathbb{Z})^{\times} \setminus (\mathbb{Z}/q\mathbb{Z})^{\times 2}$ , with $c_q$ the number of real Dirichlet characters mod $q$ divided by $2 \varphi(q)$ . A new formulation of Chebyshev’s bias was given by the asymptotic at $s = 1/2$ . When $s = 0$ , Littlewood (1914) proved that both $\pi(x) - \text{Li}(x)$ and $\pi(x, q, a) - \pi(x, q, b)$ are unbounded changing their signs infinitely many times. But the results for $0 < s < 1/2$ were unknown. In this talk, after proving the weighted prime number theorem $\pi_s(x) \sim \text{Li}(x^{1-s})$ as $x \to \infty$ , we show that both $\pi_s(x) - \text{Li}(x^{1-s})$ and $\pi_s(x, q, a) - \pi_s(x, q, b)$ are also unbounded changing their signs infinitely many times if $0 < s < 1/2$ . This is a generalization of Littlewood’s theorem, and justifies our formulation of Chebyshev’s bias which refers to the exact weight at $s = 1/2$ . This is joint work with Koji Shimada.

17 February 2025 at 15:00 in B.302
Speaker: Robert Wilms (Caen)
Title: An algebraic approach to (arithmetic) intersection theory

Abstract: Intersection theory is a fundamental tool that provides a good measure of complexity, such as the degree of an algebraic variety or the height in Diophantine geometry, while being very compatible with most geometric constructions. In this work-in-progress talk I will present a new intersection theory of norms on rings using purely algebraic means. It provides a generalisation and at the same time a completely unified formulation of the geometric and arithmetic intersection theory of line bundles. It also provides potential applications in other areas of mathematics, such as discrete geometry or the theory of modular forms, which I will also discuss.

10 February 2025 at 15:00 in B3.02
Speaker: Jessica Alessandri (Bath)
Title: TBA

Abstract: TBA

03 February 2025 at 15:00 in B3.02
Speaker: Tim Dokchitser (Bristol)
Title: TBA

Abstract: TBA

27 January 2025 at 15:00 in B3.01
Speaker: Abhishek Saha (QMUL)
Title: Holomorphic QUE for Siegel cusp forms: the case of Saito-Kurokawa lifts

Abstract: The Quantum Unique Ergodicity (QUE) conjecture was proved in the classical case of Maass forms on the upper-half plane by Lindenstrauss and Soundararajan. The analogous mass equidistribution statement for holomorphic cusp forms in the weight aspect is a theorem due to Holowinsky and Soundararajan. In this talk, I will discuss some recent joint work with Jesse Jaasaari and Steve Lester on the higher rank analogue of the result of Holowinsky and Soundararajan for the case of holomorphic Siegel cusp forms. Our main result establishes mass equidistribution for Saito-Kurokawa lifts (which are special types of Siegel cusp forms of degree 2) assuming the Generalized Riemann Hypothesis (GRH) . We also show that this implies the equidistribution of zero divisors of Saito-Kurokawa lifts. Time permitting, I will say a few words about generalisation to higher dimensions.

20 January 2025 at 15:00 in B3.02
Speaker: Robin Bartlett (Glasgow)
Title: TBA

Abstract: TBA

13 January 2025 at 15:00 in B3.02
Speaker: Max Xu (Simons/Courant Institute)
Title: Nonnegative partial sums of quadratic characters

Abstract: In 1911, Fekete proposed the problem of studying how likely a Fekete polynomial has no real zeros in [0,1]. A related question by Polya is to ask how likely a random quadratic character always has nonnegative partial sums. Notably, the work of Baker and Montgomery in 1989 qualitatively showed both events are rare, in the sense that the asymptotic density is zero. In a joint work in progress with Angelo and Soundararajan, we give a quantitative upper bound which is (we believe) close to the truth, and in fact we believe there are plenty of such polynomials and characters.

06 January 2025 at 15:00 in B3.02
Speaker: Sean Eberhard (Warwick)
Title: TBA

Abstract: TBA

02 December 2024 at 15:00 in B3.02
Speaker: Nikolaos Diamantis (Nottingham)
Title: L-series of half-integral weight cusp forms and an analogue of the period polynomial

Abstract: We construct a polynomial expressed in terms of values of the L-series attached to a half-integral weight cusp form. This polynomial can be thought of as an analogue of the classical period polynomial since it also satisfies certain "period relations". We show how it induces a lift of half-integral weight cusp forms to integral weight forms which is compatible with the L-series of the respective forms. This lift is explicit thanks to a result by Pasol and Popa. (Joint work with Branch, Raji and Rolen).

25 November 2024 at 15:00 in B302
Speaker: Elvira Lupoian (Imperial)
Title: Runge’s Method and Integral Points on Modular Curves

Abstract: The study of integral points on curves dates to Siegel’s theorem in 1929, which has been historically studied by many due to its connections to Mordell’s conjecture (Faltings’ theorem) . More recently, Bilu and Parent were able to prove Serre’s uniformity conjecture in the split Cartan case by studying integral points on the corresponding modular curve. Their proof relies on efficiently determining the integral points using the so -called Runge’s method. In this talk, we review this method and discuss how the work of Bilu and Parent can be adapted to efficiently bounds heights of integral points on certain covers of the classical modular curve $X_{0} \left( p \right)$ .

18 November 2024 at 15:00 in B3.02
Speaker: Emma Bailey (Bristol)
Title: Large deviations of Selberg’s CLT: upper and lower bounds

Abstract: Selberg’s CLT informs us that the logarithm of the Riemann zeta function evaluated on the critical line behaves as a complex Gaussian. It is natural, therefore, to study how far this Gaussianity persists. This talk will present conditional and unconditional results on atypically large values, and is joint with Louis-Pierre Arguin and Asher Roberts.

11 November 2024 at 15:00 in B.302, Zeeman building
Speaker: Chris Daw (University of Reading)
Title: Large Galois orbits under multiplicative degeneration

Abstract: The Pila-Zannier strategy is a powerful technique for proving results in unlikely intersections. In this talk, I will recall the Zilber-Pink conjecture for Shimura varieties and describe how Pila-Zannier works in this setting. I will highlight the most difficult outstanding obstacle to implementing the strategy — the so-called Large Galois Orbits conjecture — and I will explain recent progress towards this conjecture, building on the works of André and Bombieri. This is joint with Martin Orr (Manchester).

04 November 2024 at 15:00 in B.302, Zeeman building
Speaker: Andrei Yafaev (University College London)
Title: TBA

Abstract: TBA

28 October 2024 at 15:00 in B3.02
Speaker: Akshat Mudgal (Warwick)
Title: Recent progress towards the sum–product conjecture and related problems

Abstract: An important open problem in combinatorial number theory is the Erdős–Szemerédi sum–product conjecture, which suggests that for any positive integers s, N, and for any set A of N integers, either there are many s-fold sums of the form $a_1 + … + a_s$ or there are many s-fold products of the form $a_1\dots a_s$ . While this remains wide open, various generalisations of this problem have been considered more recently, including the question of finding optimal variations of the so-called low energy decompositions. In this talk, I will outline some recent progress towards the above questions, as well as highlight how these connect very naturally to other key conjectures in additive combinatorics.

21 October 2024 at 15:00 in B302
Speaker: Tobias Berger (University of Sheffield)
Title: Pseudomodularity of residually reducible Galois representations

Abstract: After a survey of previous work I will present new results on pseudomodularity of residually reducible Galois representations with 3 residual pieces. I will discuss applications to proving modularity of Galois representations arising from abelian surfaces and Picard curves. This is joint work with Krzysztof Klosin (CUNY).

14 October 2024 at 15:00 in B3.02
Speaker: Maleeha Khawaja (Sheffield)
Title: Galois groups of low degree points on curves

Abstract: Whilst the study of low degree algebraic points on curves is an active area of research, there has been little emphasis on the Galois-theoretic description of these points. In this talk, we focus on the behaviour of low degree points whose Galois group is primitive. Furthermore, we shall see that the behaviour changes if the degree is large (with respect to the genus of the curve). This talk is based on joint work with Frazer Jarvis (Sheffield) and Samir Siksek (Warwick).

07 October 2024 at 15:00 in B.302, Zeeman building
Speaker: Han Yu (University of Warwick)
Title: TBA

Abstract: TBA

30 September 2024 at 15:00 in B3.02
Speaker: Tim Browning (IST Austria)
Title: Pairs of commuting matrices

Abstract: I'll discuss commuting varieties and a new upper bound for the number of pairs of commuting $n \times n$ matrices with integer entries and height at most $T$ , as $T \to \infty$ . Our approach uses Fourier analysis and mod $p$ information, together with a result about the flatness of the commutator Lie bracket, which we also solve. This is joint work with Will Sawin and Victor Wang.