Program & Abstracts

The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety X into the set of its adelic points.

The Brauer-Manin pairing cuts out a subset of the adelic points, called the Brauer-Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer-Manin set is empty then we say there's a Brauer-Manin obstruction to the existence of rational points on X.

Computing the Brauer-Manin pairing involves evaluating elements of the Brauer group of X at local points. If an element of the Brauer group has order coprime to p, then its evaluation at a p-adic point factors via reduction of the point modulo p. For p-torsion elements this is no longer the case: in order to compute the evaluation map one must know the point to a higher p-adic precision. Classifying p-torsion Brauer group elements according to the precision required to evaluate them at p-adic points gives a filtration which we describe using work of Bloch and Kato.

Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer-Manin obstruction. This is joint work with Martin Bright.

Slides available here.

Let K be a discretely valued field with ring of integers R. To a d-by-d matrix M with integral coefficients one can associate an R-module, in K^{d\times d}, and a polytope, in the Euclidean space of dimension d-1. We will look at the interplay between these two objects, from the point of view of tropical geometry and building on work of Plesken and Zassenhaus.

This is joint work with Y. El Maazouz, M. A. Hahn, G. Nebe, and B. Sturmfels.

In this talk we will first review the classical criteria to determine the (stable) reduction type of elliptic curves (Tate) and of genus 2 curves (Liu) in terms of the valuations of some particular combinations of their invariants. We will also revisit the theory of cluster pictures to determine the reduction type of hyperelliptic curves (Dokchitser's et al.). Via Mumford theta constants and Takase and Tomae's formulas we will be able to read the cluster picture information by looking at the valuations of some (à la Tsuyumine) invariants in the genus 3 case. We will also discuss the possible generalization of this strategy for any genus and some related open questions.

For an elliptic curve defined over a number field, we give an explicit description its group of adelic points.

It turns out that the structure of this topological group depends only on the degree of the number field and the Galois representation associated to

its torsion points. This is a result that grew out of the 2015 ALGANT thesis of Athanasios Angelakis.

Pick an integer n. Consider a natural family of objects, such that each object X in the family has an L-function L(X,s). If we assume that the collection of special values L*(X,n) is bounded, does it imply that the family of objects is finite?

We will first explain why we consider this question, in link with Kato's heights of mixed motives, and give two recent results.

This is joint work with Riccardo Pengo.

In 2020, Ichiro Schimada and Davide Cesare Veniani give a formula to compute the number of isomorphism classes of Enriques surfaces that are doubly covered by a K3 surface with a given transcendental lattice. In the same paper, they apply this formula to singular K3 surfaces.

In this talk, we apply the formula to K3 surfaces X with Picard number 19 and |det T_X |< 16, where T_X is the transcendental lattice of X.

We compute the number of and study the isomorphism classes of Enriques surfaces that are doubly covered by X.

This is joint work with D. Veniani.

Consider the integral model S of a Shimura variety with good reduction, in mixed characteristic 0-p. Caraiani and Scholze construct a perfectoid cover of the generic fiber of S and the so called Hodge-Tate period map, with target a suitable flag variety. In this talk I will compare the pull-back to this perfectoid cover of two stratifications. The first is the Ekedhal-Oort stratification on the mod p special fiber of S. The second is the fine Deligne-Lusztig stratification on the mod p special fiber of the flag variety.

Congruences between modular forms play a crucial role in understanding links between geometry and arithmetic: cornerstone example of this is the proof of Serre's modularity conjecture by Khare and Wintenberger.

Congruences of Galois representations govern many kinds of representations of the absolute Galois group of number fields. Even though our understanding is improving, many aspects remain very mysterious, some are theoretically approachable, many are not; and amongst the latter, some allow numerical studies to reveal first insights.

In this talk I will introduce congruence graphs, which are graphs encoding congruence relations between classical newforms (joint work with Vandita Patel). Then I will explain how to construct analogous graphs for congruences of Galois representations, and I will motivate the study of these objects trough questions regarding Hecke algebras and Atkin-Lehner operators.