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

• Autores: Gerard Freixas i Montplet
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 63, Fasc. 3, 2012, págs. 243-259
• Idioma: inglés
• DOI: 10.1007/s13348-012-0061-4
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• Resumen

  • 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.