The $L$-function of the surface parametrizing cuboids
Madoka Horie, Takuya Yamauchi
TL;DR
The paper determines the $L$-function of the cuboid parametrizing surface $S$ and elucidates the structure of its Picard group ${\rm Pic}(S_{\overline{\mathbb{Q}}})$ as a $G_Q$-module by leveraging Stoll and Testa's geometric description of the compactification $\bar S$. It embeds the problem in the realm of modular forms by identifying $\bar S$ with a quotient of $X(8)\times X(8)$ and expressing the middle étale cohomology in terms of cusp forms with CM by $\mathbb{Q}(i)$ and $\mathbb{Q}(\sqrt{2})$, yielding explicit $L$-factors such as $L(s, h_{16})$, $L(s, h_{32})$, and $L(s, h_{8})$, along with a twisted Tate component. The Picard lattice is shown to have rank $64$, generated by 34 divisors defined over $\mathbb{Q}$, 26 over $\mathbb{Q}(i)$, 1 over $\mathbb{Q}(\sqrt{-2})$, and 3 over $\mathbb{Q}(\sqrt{2})$, providing a complete description of the algebraic and arithmetic structure of $S$ and its twist $\bar S$. The results connect the geometry of rational cuboids to automorphic forms, enabling explicit factorization of the relevant $L$-functions and offering a foundation for potential extensions to diagonal quotient surfaces.
Abstract
In this note, we compute the $L$-function of the projective smooth surface $S$ over $\mathbb{Q}$ that parametrizes cuboids whose geometric properties are studied in detailby Stoll and Testa. As a byproduct, we completely determine the structure of ${\rm Pic}(S_{\overline{\mathbb{Q}}})$ as a ${\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-module.