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

The modularity of an elliptic curve $E/\mathbb Q$ 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 $X_0(N)$. 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 $1$.