Séminaire TN

TBA

TBA

Triangular modular curves are a generalization of modular curves and arise from quotients of the complex upper half-plane by congruence subgroups of hyperbolic triangle groups. They are connected to Darmon’s program for determining the rational points on generalized Fermat equations. In this talk, we will focus on their arithmetic and will classify the Borel-kind triangular modular curves by their genera. This is joint work with John Voight.

The l-isogeny graph, where vertices are supersingular elliptic curves and

edges are isogenies of prime degree l, is a fundamental building block in isogeny-based

cryptography. Understanding the expansion properties of these graphs is crucial for

assessing the security of the underlying cryptographic problems. In dimension 1 (in the

elliptic curve case), we know these graphs are Ramanujan, exhibiting optimal

expansion. But what happens to this landscape when we move to higher dimensions?



In this talk, we will explore the expansion properties of higher-dimensional isogeny

graphs, both with and without level structure. While we show that these higher-

dimensional analogues lose their Ramanujan property, we will demonstrate that they still

maintain remarkably strong expansion. Finally, we will discuss the practical applications

of these findings, specifically focusing on how they allow us to remove the heuristics from

the Costello-Smith algorithm for solving the dimension-2 isogeny problem.

This is based on joint work with M. Corte-Real Santos, O. Taïbi, and B. Wesolowski.