The arithmetic volume of the moduli space of abelian surfaces

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, Kühn 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 $g=2$.