Modular elliptic curves and hyperbolic uniformization

Abstract

In an article published a few years before the modularity of elliptic curves over $\Q$ was proved, Mazur \cite{maz} looked at modularity as a purely complex analytic phenomenon, defining a notion of an elliptic curve over $\Q$ having a hyperbolic uniformisation of arithmetic type. Such an elliptic curve (of conductor $N$, say) is necessarily geometrically modular, i.e. a quotient of the jacobian of the modular curve $X_0(N)$, by a morphism defined over $\Q$. We extend these ideas to elliptic curves over totally real fields of odd degree, using Shimura curves for quaternion algebras split at all finite places and one real place. In particular, we prove that the existence of a hyperbolic uniformisation of arithmetic type would imply geometric modularity.