The $L$-function of the surface parametrizing cuboids

TL;DR

This paper computes the $L$-function of a smooth surface over $Q$ that parametrizes cuboids, providing insights into its geometric and arithmetic properties, including the Picard group structure.

Contribution

It explicitly determines the $L$-function of the surface and describes the Galois module structure of its Picard group, advancing understanding of the surface's arithmetic geometry.

Findings

Computed the $L$-function of the surface

Determined the Picard group structure as a Galois module

Enhanced understanding of the surface's geometric properties

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 detail by 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.