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.
