An arithmetic Riemann-Roch theorem for pointed stable curves

TL;DR

This paper establishes an arithmetic Riemann-Roch theorem specifically for pointed stable curves and explores its implications for the Selberg zeta function of certain modular curves, advancing the understanding of arithmetic geometry.

Contribution

It introduces a new arithmetic Riemann-Roch theorem tailored for pointed stable curves and applies it to analyze Selberg zeta functions of specific modular curves.

Findings

Derived consequences for the Selberg zeta function of $Y_{1}(p)$ and $Y_{0}(p)$

Established an arithmetic Riemann-Roch theorem for pointed stable curves

Connected arithmetic geometry with spectral properties of modular curves

Abstract

We prove an arithmetic Riemann-Roch theorem for pointed stable curves. We derive consequences for the Selberg zeta function of an open modular curve $Y_{1}(p)$ (resp. $Y_{0}(p)$), for a prime number $p\geq 11$ (resp. congruent to 11 modulo 12).