The geometry and arithmetic of a Calabi-Yau Siegel threefold

TL;DR

This paper explores the structure and geometry of a specific Calabi-Yau threefold derived from a modular variety, employing algebraic and geometric methods to analyze its Hodge numbers and modular form ring.

Contribution

It provides a detailed analysis of a Calabi-Yau Siegel threefold, including two methods for computing Hodge numbers and understanding its modular form ring structure.

Findings

Explicit description of the modular form ring for the variety

Two methods for computing Hodge numbers demonstrated

Determination of the Picard group and Euler characteristic

Abstract

In this paper we treat in details a modular variety $\cal Y$ that has a Calabi-Yau model, $\tilde{\cal Y}$. We shall describe the structure of the ring of modular forms and its geometry. We shall illustrate two different methods of producing the Hodge numbers. The first uses the definition of $\cal Y$ as the quotient of another known Calabi-Yau variety. In this case we will get the Hodge numbers considering the action of the group on a crepant resolution $\tilde{\cal X}$ of $\cal X$. The second, purely algebraic geometric, uses the equations derived from the ring of modular forms and is based on determining explicitly the Calabi-Yau model $\tilde{\cal Y}$ and computing the Picard group and the Euler characteristic.