London-Paris number theory seminar: Talk titles and abstracts

Lassina Dembélé, King’s College London

Title: Semistable abelian varieties with good reduction outside 73.

Abstract: In this talk, we give a classification of semistable abelian varieties over the rationals, with good reduction outside 73. The classification assumes GRH.

Tim Dokchitser, University of Bristol

Title: A classification for reduction types of curves

Abstract: The primary invariant for a family of curves is the combinatorial description of `bad' fibers. When the curves are elliptic, the classification of possible geometric configurations (`reduction types') is due to Kodaira and Neron, in genus 2 to Namikawa-Ueno, and in genus 3 to Ashikaga-Ishizaka. In this talk, I would like to describe a possible classification for curves of arbitrary genus.

Elisa Lorenzo García, Université de Neuchâtel and Université de Rennes 1

Title: Reduction of Plane Quartics and Cayley Octads

(j.w.w. Raymond van Bommel, Jordan Docking, Vladimir Dokchitser and Reynald Lercier)

Abstract: We will first give a conjectural characterization, based on numerical experimentation, of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This will result in p-adic criteria that efficiently give the stable reduction type amongst the 42 possible types, and whether the reduction is hyperelliptic or not. These criteria are in the vein of the machinery of "cluster pictures" for hyperelliptic curves. Later, we will discuss how to read this characterisation from an invariant theory point of view and some ideas on how to prove the conjecture.

Chloe Martindale, University of Bristol

Title: Making and breaking post-quantum cryptography from elliptic curves

Abstract: Most of the public-key cryptography in use today relies on the hardness of either factoring or the discrete logarithm problem in a specially chosen abelian group. Here "hard" does not mean mathematically impossible but that the best known algorithm to solve the problem has complexity (sub-)exponential in the size of the input. However, once scalable quantum computers become a reality, both factoring and the discrete logarithm problem will no longer be hard problems, due to Shor's polynomial-time quantum algorithm to solve both problems. Post-quantum cryptography is about designing new cryptographic primitives based on different hard problems in mathematics for which there is no known polynomial-time classical or quantum algorithm. In this talk we will show how to design post-quantum cryptographic primitives from the hard problem of, given two elliptic curves over a large finite field, find and compute and isogeny between them (if it exists). We will then discuss recent work giving an attack on one of these primitives, Supersingular Isogeny Diffie-Hellman (SIDH). This is joint work with Luciano Maino, Lorenz Panny, Giacomo Pope, and Benjamin Wesolowski.

Pascal Molin, Institut de Mathématiques de Jussieu

Title: Numerical computations around Harris-Venkatesh conjectures

Abstract: The recent conjectures of Harris-Venkatesh predict Stark-type relations for weight 1 modular forms and their associated Artin representation. I will present some experiments with modular forms of exotic type and the computational challenges that remain.

Aurel Page, Institut de Mathématiques de Bordeaux

Title: What can we compute in 2024? Overview and open problems in computational number theory

Abstract: I will give a survey of some topics in computational number theory, focusing on concretely accessible examples, interactions with theoretical progress, and open problems.