Adam Logan, University of Waterloo, TIMC and Carleton University
Computational exploration of Hilbert modular surfaces
There are at least two viewpoints on the modularity of elliptic curves over the rationals: it can be seen either as an analytic and representation-theoretic statement that the L-function of a curve is associated to a modular form, or as a geometric statement that the curve is a quotient of a modular curve. It is not clear that these remain equivalent for elliptic curves over number fields. For elliptic curves over real quadratic fields, analytic modularity is now known, and a form of geometric modularity was conjectured 40 years ago by Oda. Recent advances in the computation of rings of Hilbert modular forms have made it possible to verify the geometric modularity conjecture in special cases. In this talk I will describe my work in this direction, including some interesting auxiliary algebraic surfaces that arise in the course of the computations.
MC 5417