London Junior Number Theory Seminar

 Talks

Geometric Galois module theory and epsilon constants

09 December 2025, Eva Brenner (LMU Munich)

Abstract: There is a deep connection between Galois module structure and epsilon constants that appear in the functional equations of L-functions. After briefly reviewing classical results for the ring of integers of a number field, we will turn to the case of arithmetic varieties with a tame action of a finite group. In this context, epsilon constants of symplectic characters determine the Galois module structure coming from de Rham cohomology. Recovering epsilon constants from the Galois module structure requires additional data, in the form of a Hermitian form or a metric. I will describe two constructions, coming respectively from Serre duality and from Quillen metrics, and discuss how they relate.

From quantum entanglement to the theta correspondence

25 November 2025, Yiannis Fam (KCL)

Abstract: Albert Einstein once described entanglement as "spooky action at a distance". But what if that distance was non-archimedean? In this off-the-rails talk I will give a brief and naive introduction to quantum mechanics and describe how one construction of entangled particles plays a spooky role in the representation theory of p-adic groups.

Chebotarev geodesic theorem

11 November 2025, Alberto Acosta Reche (UCL)

Abstract: 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.

Minimal motive for double zeta values

28 October 2025, Kenza Memlouk (Université de Strasbourg)

Abstract: In this talk, we consider multiple zeta values, which are periods of certain motives. Given a multiple zeta value ζ, there exists a unique minimal motive so that ζ is a period of this motive. In general, this motive is very difficult to compute. In the specific case of a double zeta value ζ(a,b), we can compute such a minimal motive M(a,b). This computation allows us to formulate conjectures about transcendence of double zeta values. We will give the Tannakian group associated to M(a,b) and discuss the differences between the cases of simple zeta values and double zeta values.

Recovering reduction types of elliptic curves through torsion

14 October 2025, Naina Praveen (UCL)

Abstract: The classification of an elliptic curve's reduction at a prime p yields interesting arithmetic invariants. Classically, one can determine this reduction type via Tate's algorithm, a method that depends on explicit Weierstrass models. A more intrinsic approach is to use representation theory — the Néron-Ogg-Shafarevich criterion allows for a classification of potentially good reduction types for primes p>3. In this talk, I shall introduce another method that extends these results by incorporating the geometric configuration of the l-torsion points, specifically their p-adic distances. This approach extends the full classification to the case of p=3. I will then present a rough classification for p=2 and walk through the problems/examples that prevent a complete determination in that case, along with a sketch of the proof. If time permits, I might introduce how the above techniques could be generalised to genus 2 curves.