Sam Frengley

Congruences and the geometry of Humbert surfaces

Pairs of elliptic curves are said to be N-congruent if their N-torsion subgroups are isomorphic as Galois modules. Such elliptic curves arise, for example, when one studies splittings/gluings of abelian surfaces. Pairs of N-congruent elliptic curves are parametrised by certain Hilbert modular surfaces and the geometry of these surfaces was studied by Kani--Shanz.

It is a conjecture of Frey--Mazur that, over the rational numbers, no non-isogenous N-congruent elliptic curves exist for sufficiently large integers N, but a precise bound is yet to be conjectured in general. I will discuss progress towards refining this conjecture, firstly by computing such Hilbert modular surfaces when N=12,14,15 and secondly by studying the birational geometry of certain quotients, known as Humbert
surfaces.