An explicit formula of the normalized Mumford form

TL;DR

This paper presents an explicit, infinite product formula for the normalized Mumford form on the moduli space of algebraic curves, generalizing the Ramanujan delta function to higher genus cases.

Contribution

It provides the first explicit formula expressing the second tautological line bundle via the Hodge line bundle for all genera, enabling computable power series with integral coefficients.

Findings

Explicit infinite product formula for the normalized Mumford form

Universal power series expression with integral coefficients

Generalization of Ramanujan delta function to higher genus

Abstract

We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented by an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. By this formula, we have a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves.