Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda–Hamahata conjecture

The modularity of an elliptic curve E/Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E/{\mathbb Q}$$\end{document} can be expressed either as an analytic statement that the L-function is the Mellin transform of a modular form, or as a geometric statement that E is a quotient of a modular curve X0(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0(N)$$\end{document}. For elliptic curves over number fields these notions diverge; a conjecture of Hamahata asserts that for every elliptic curve E over a totally real number field there is a correspondence between a Hilbert modular variety and the product of the conjugates of E. In this paper we prove the conjecture by explicit computation for many cases where E is defined over a real quadratic field and the geometric genus of the Hilbert modular variety is 1.