Analytic Torsion and Automorphic Forms with Infinite Products

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(1) In 1998, we introduced an invariant of a K3 surface with anti-symplectic involution : By fixing a Ricci-flat Kaehler metric on a K3 surface, which is invariant under the involution, the notion of the equivariant analytic torsion of the K3 surface with involution and of the analytic torsion of the fixed curves make sense. Then, the product of these two quantities is our invariant. In these two years, it has become possible to define the invariant without using Yau's theorem, the existence of Ricci-flat Kaehier metrics on a K3 surface. Indeed, by adding certain factor of Bott-Chern class to the previous definition, onecan obtain the same invariant without assuming the Ricci-flatness of the metric. This invariant is represented by an automorphic form on the moduli space. Before this progress, it was inevitable to study the degenerating behavior of Ricci-flat metrics, which made our proof hard to read. The fact that the invariant is independent of the choice of a metric, reduces the study of its degenerating behavior to that of Bott-Chern classes. It is much easier to understand the degenerations of Bott-Chern classes than that of Einstein metrics.(2) It was known before that the analytic torsion of curves of genus 1 (resp. 2) is represented by a certain Siegel modular form. By a jointwork with Shu KAWAGUCHI (Kyoto Univ.), we extend this fact to curves of genus 3. More precisely, their Quillen metric is represented by a certain Siegel modular form. The key fact is that every non-hyperelliptic curve of genus 3 is a hyperplane section of a Kummer's quartic. However, that realization depends on the choice of an unramified double covering of the curve.