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arXiv · 2411.08269

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

Adam Logan
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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 $1$.

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Adam Logan. 2024-11-13. Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture. https://arxiv.org/abs/2411.08269

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