A product formula of Mumford forms, and the rationality of Ruelle zeta values for Schottky groups

TL;DR

This paper derives an explicit product formula for Mumford forms using infinite products similar to Selberg zeta functions for Schottky groups, extending known genus 1 results, and applies it to analyze the rationality of Ruelle zeta values.

Contribution

It provides a new explicit formula for Mumford forms in higher genus using infinite products, extending classical genus 1 results, and explores implications for Ruelle zeta value rationality.

Findings

Explicit product formula for Mumford forms in positive genus

Extension of genus 1 formula involving Ramanujan's Delta function

Application to the rationality of Ruelle zeta values for Schottky groups

Abstract

In any positive genus case, we show an explicit formula of the Mumford forms expressed by infinite products like Selberg type zeta values for Schottky groups. This result is considered as an extension of the formula in terms of Ramanujan's Delta function in the genus 1 case, and is especially applied to studying the rationality of Ruelle zeta values for Schottky groups.