Number Theory

Number Theory Seminar

Schedule 2025/26

Abstracts 2025/26

Sara Checcoli, A little bit of little points generated by torsion

The height is a non-negative real-valued function that measures the arithmetic complexity of an algebraic number. While numbers of minimal height are well understood, many questions remain about numbers of small but non-zero height. One such question is whether a given algebraic extension K of the rationals contains only finitely many elements of bounded height, in which case, following Bombieri and Zannier, K is said to have the Northcott property (N). While this holds for number fields, the situation is more subtle for infinite extensions of the rationals. For example, the maximal abelian extension of the rationals does not have (N), but it follows from a result of Bombieri and Zannier that its subextensions with Galois group of bounded exponent do. In this talk, after giving an overview on the subject, I will present joint work with Gabriel Dill establishing, in particular, a similar result for fields generated by torsion points of abelian varieties over number fields.

Pengcheng Zhang, Meromorphic modular forms and symmetric powers of elliptic curves

We will discuss the link between meromorphic modular forms and symmetric powers of elliptic curves through the p-adic behaviors of the former. In particular, we will discuss the heuristical connection as well as justify the heuristics through numerical observations.

Julien Soumier, From Post-Quantum Cryptography to Isomorphisms of Abelian Varieties

In this talk we present an algorithm, polynomial under GRH, for computing isomorphisms between products of supersingular elliptic curves with known endomorphism rings. Our work can be seen as an effective version of a theorem of Delign, Ogus and Shioda. It is motivated by the growing interest in the computation of abelian varieties for post-quantum cryptography. In order to present our work, we will begin by reviewing the mathematical objects involved and explaining why they have become relevant for cryptographic purposes. Then we will describe our approach that leverages the Deuring correspondence, enabling us to reformulate computational isogeny problems into algebraic problems in quaternion algebras. We will conclude by a brief overview of the perspectives it offers.

Steven Charlton, Dedekind Zeta Values, Zagier's Polylogarithm Conjecture and depth reductions for multiple polylogarithms

Dirichlet's famous Class Number Formula expresses the residue of the Dedekind zeta function ζ_F(s) of a number field F in terms of important arithmetic information about the number field, including, in particular, the regulator which is a determinant involving logarithms of units of O_F. Zagier's Polylogarithm Conjecture extends this to express values of ζ_F(n), n = 2, 3, 4, ... in terms of suitable combinations of n-logarithms, giving 'higher' units. Goncharov outlined a strategy to tackle Zagier's Polylogarithm Conjecture, by understanding first the maximal depth multiple polylogarithms and the geometrically defined Grassmannian polylogarithm, before attempting to successively reduce the depth, to reach the classical n-logarithm. I will outline some of the history of Zagier's Polylogarithm Conjecture, and of Goncharov's Programme. I will then briefly discuss some recent developments, including work by Goncharov-Rudenko proving ζ_F(4); by C-Gangl-Radchenko on formulas for Grassmannian polylogarithms making this explicit, and some separate work by Matveiakin-Rudenko and by myself which establishes new reductions in weight 6.

Jessica Alessandrì, Fields generated by torsion points of elliptic curves

In the study of elliptic curves, division fields, that are fields generated by torsion points of the curves, play an important role. For example in Galois representation, modularity and the proof of the Mordell-Weil theorem. In this talk we will present some results on torsion points and division fields. In particular, we will see how to compute torsion points, their properties and how to find a "small" set of generators. We give some results for 7-division fields of CM elliptic curves and some applications of them, based on a joint work with Prof. L. Paladino.

Álvaro Lozano-Robledo, An introduction to adelic Galois representations attached to elliptic curves, and entanglements

In this talk we will describe how to construct the adelic Galois representation attached to an elliptic curve, and we will motivate why one would want to construct such a thing, and what kind of results one can show once the image of the Galois representation is known. In particular, we will discuss the problem of entanglements of division fields of elliptic curves, and discuss some of the recent results on the subject.

Ivan Novak, Quadratic points on \(X_0(n)\) and \(\mathbb Q\)-curves

For a positive integer \(n\), the modular curve \(X_0(n)\) parametrizes elliptic curves with a cyclic isogeny of degree \(n\). In this talk, we discuss quadratic points on the curves \(X_0(n)\). We first recall known results on rational points and then describe current progress on the program of determining quadratic points. Specifically, all quadratic points on all \(X_0(n)\) for \(n\leq 100\) have been determined. The study of quadratic points on \(X_0(n)\) naturally leads to quadratic \(\mathbb Q\)-curves, which are elliptic curves isogenous to their Galois conjugates. We discuss the general theory of \(\mathbb Q\)-curves, and then present results on the classification of their torsion subgroups over prime degree number fields. Most of the original results presented in the talk are joint work with Filip Najman.

Past Seminar Editions

• Number Theory Seminar 2024/25
• Number Theory Seminar 2023/24
• Number Theory Seminar 2022/23
• Number Theory Seminar 2021/22
• Number Theory Seminar 2020/21