An Arakelov tautological boundary divisor on ${{\overline{\mathcal{M}}_{1,1}}}$

Abstract

We define a natural singular hermitian metric ${\|\cdot\|_{s}}$ (s > 0) on the boundary divisor ${{\delta=\mathcal{O}(\partial\mathcal{M}_{1,1})}}$ of the moduli stack of 1-pointed stable curves of genus 1, ${{\overline{\mathcal{M}}_{1,1}}}$ . For s > 3/2 we prove that ${\|\cdot\|_{s}}$ is a log-singular hermitian metric in the sense of Burgos–Kramer–Kühn, with singularities along ${{\partial\mathcal{M}_{1,1}}}$ . We compute the arithmetic intersection number of ${(\delta,\|\cdot\|_{s})}$ with the first tautological hermitian line bundle ${\overline{\kappa}_{1,1}}$ on ${{\overline{\mathcal{M}}_{1,1}.}}$ The result involves the special values ${{\zeta^{\prime}(-1), \zeta^{\prime}(-2)}}$ and ${{\zeta(2, s)}}$, where ${\zeta(s)}$ is Riemann’s zeta function and ${\zeta(\sigma,s)}$ is Hurwitz’ zeta function.