An Arakelov tautological boundary divisor on \({\overline{\mathcal{M}}_{1,1}}\)
type: Item
instance of: scholarly article
title: An Arakelov tautological boundary divisor on \({\overline{\mathcal{M}}_{1,1}}\)
zbMATH Open document ID: 1273.14047
author: Gerard Freixas i Montplet
published in: Collectanea Mathematica
publication date: 2013-06-13T00:00:00Z
review text: 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.
reviewed by: Shun Tang
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 6175348
zbMATH Keywords: Arakelovian boundary divisor
zbMATH Keywords: moduli stack
zbMATH Keywords: 1-pointed stable curves of genus 1
zbMATH Keywords: arithmetic intersection number
zbMATH Keywords: special values of zeta function
MaRDI profile type: Publication
full work available at URL: s13348-012-0061-4
OpenAlex ID: W2028133767
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DOI: 10.1007/S13348-012-0061-4
description: scientific article; zbMATH DE number 6175348