Resumen de An Arakelov tautological boundary divisor on ⎯⎯⎯⎯⎯⎯⎯1,1

Gerard Freixas i Montplet

• 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.