# Effective upper bound of analytic torsion under Arakelov metric

## TL;DR

This paper provides an asymptotic upper bound for the analytic torsion of Riemann surfaces under Arakelov metric, contributing to the understanding of spectral invariants in Arakelov geometry.

## Contribution

It offers the first effective asymptotic estimate of analytic torsion under Arakelov metric, advancing the study of spectral invariants in arithmetic geometry.

## Key findings

- Logarithm of analytic torsion is asymptotically bounded by genus g for g > 1.
- Provides evidence for potential bounds on cohomology dimensions in Arakelov theory.
- Establishes a link between spectral invariants and arithmetic surface cohomology.

## Abstract

Given a choice of metric on the Riemann surface, the regularized determinant of Laplacian (analytic torsion) is defined via the complex power of elliptic operators: $$ \det(\Delta)=\exp(-\zeta'(0)) $$ In this paper we gave an asymptotic effective estimate of analytic torsion under Arakelov metric. In particular, after taking the logarithm it is asymptotically upper bounded by $g$ for $g>1$. The construction of a cohomology theory for arithmetic surfaces in Arakelov theory has long been an open problem. In particular, it is not known if $h^{1}(X,L)\ge 0$. We view this as an indirect piece of evidence that if such a cohomology theory exists, the $h^{1}$ term may be effectively estimated.