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

TL;DR

This paper proves the Hamahata conjecture for many elliptic curves over real quadratic fields by explicit computation of Hilbert modular forms and surfaces, linking geometric and arithmetic properties.

Contribution

It provides the first explicit computational verification of the Hamahata conjecture for a broad class of elliptic curves over real quadratic fields.

Findings

Confirmed the conjecture for numerous cases over real quadratic fields

Computed Hilbert modular forms and surfaces explicitly

Established connections between elliptic curves and Hilbert modular varieties

Abstract

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 $1$.