Analytic Torsion and Automorphic Forms with Infinite Products

(1)1998年，我们引入了具有反辛对合的K3曲面的一个不变量：通过在K3曲面上固定Ricci-Flat Kaehler度量，它在对合下是不变的，K3曲面具有对合的等变解析挠率和固定曲线的解析挠度的概念是有意义的。那么，这两个量的乘积就是我们的不变量。在这两年中，定义不变量已经成为可能，而不需要使用Yau定理，即K3曲面上Ricci平坦Kaehier度量的存在性。实际上，通过在以前的定义中加入Bott-Chern类的某些因子，可以在不假定度规的Ricci平坦性的情况下获得相同的不变量。这个不变量由模空间上的一个自同构形式表示。在此之前，研究Ricci-Flat度量的退化行为是不可避免的，这使得我们的证明很难读懂。不变量与度规的选择无关的事实，使对其退化行为的研究简化为Bott-Chern类的研究。与爱因斯坦度规相比，更容易理解Bott-Chern类的退化。(2)以前就知道亏格1的曲线的解析挠率。2)由某一Siegel模形式表示。通过与Shu Kawaguchi(京都大学)的合作，我们将这一事实推广到亏格3的曲线。更准确地说，它们的Quillen度量由某种Siegel模形式表示。关键的事实是，亏格3的每条非超椭圆曲线都是Kummer四次曲线的超平面。然而，这一点的实现取决于曲线的非分支双重覆盖的选择。

(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.