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 $g=2$.
Categories: math.AG math.NT