An Arakelov tautological boundary divisor on \({\overline{\mathcal{M}}_{1,1}}\)
The content of the paper under review is expressing special values of some zeta function and its derivative as the Archimedean contribution to some arithmetic intersection numbers. Precisely, the author focused 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 any positive real number \(s\), he constructed a singular metric \(\parallel\cdot\parallel_s\) on \(\delta\) as \[ \parallel1\parallel_{s,(X,p)}=\prod_\gamma\prod_{k=0}^{\infty}(1-e^{-(s+k)l(\gamma)})^{12} \] where \(1\) stands for the canonical section of \(\delta\), \((X,p)\in \mathcal{M}_{1,1}(\mathbb{C})\), and \(\gamma\) runs over non-oriented simple closed hyperbolic geodesics on \(X\backslash\{p\}\) with length \(l(\gamma)\). This metric was shown by the author to be log-singular whenever \(s>\frac{3}{2}\) with singularities along \(\partial\mathcal{M}_{1,1}\), so that the arithmetic intersection number of \((\delta,\parallel\cdot\parallel_s)\) with the first tautological hermitian line bundle \(\overline{\kappa}_{1,1}\) on \(\overline{\mathcal{M}}_{1,1}\) is well defined in the sense of Arakelov. The main result in the paper under review is that this intersection number \[ (\overline{\delta}_s,\overline{\kappa}_{1,1})=3\zeta'(-1)-12\zeta(2,s)\zeta'(-2)+\frac{5}{6}\log2-\frac{1}{2} \] where \(\zeta\) is the Riemann's zeta function and \(\zeta(\sigma,s)=\sum_{k=0}^\infty(k+s)^{-\sigma}\) is the Hurwitz's zeta function.
scientific article; zbMATH DE number 6175348
scientific article