An Arakelov tautological boundary divisor on ⎯⎯⎯⎯⎯⎯⎯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 ‖⋅‖? (s > 0) on the boundary divisor ?=(∂1,1) of the moduli stack of 1-pointed stable curves of genus 1, ⎯⎯⎯⎯⎯⎯⎯1,1 . For s > 3/2 we prove that ‖⋅‖? is a log-singular hermitian metric in the sense of Burgos–Kramer–Kühn, with singularities along ∂1,1 . We compute the arithmetic intersection number of (?,‖⋅‖?) with the first tautological hermitian line bundle ?⎯⎯⎯1,1 on ⎯⎯⎯⎯⎯⎯⎯1,1. The result involves the special values ?′(−1),?′(−2) and ?(2,?) , where ?(?) is Riemann’s zeta function and ?(?,?) is Hurwitz’ zeta function.