The arithmetic volume of the moduli space of abelian surfaces

TL;DR

This paper extends the known formula for the arithmetic volume of the moduli space of abelian varieties from dimension 1 to dimension 2, involving special values of the Riemann zeta function.

Contribution

It generalizes Kuhn's 1999 formula for the arithmetic volume from genus 1 to genus 2 abelian varieties.

Findings

Derived an explicit formula for the arithmetic volume of _2

Connected the volume to special values of the Riemann zeta function

Extended previous results from genus 1 to genus 2

Abstract

Let $\mathcal{A}_g$ denote the moduli stack of principally polarized abelian varieties of dimension $g$. The arithmetic height, or arithmetic volume, of $\overline{\mathcal{A}}_g$, is defined to be the arithmetic degree of the metrized Hodge bundle $\overline{\omega}_g$ on $\overline{\mathcal{A}}_g$. In 1999, K\"uhn proved a formula for the arithmetic volume of $\overline{\mathcal{A}}_1$ in terms of special values of the Riemann zeta function. In this article, we generalize his result to the case $g=2$.