Arithmetic Riemann-Roch and Hilbert-Samuel formulae for pointed stable curves

TL;DR

This paper establishes arithmetic Riemann-Roch and Hilbert-Samuel formulas for pointed stable curves, providing new tools for understanding their geometric and arithmetic properties.

Contribution

It introduces novel arithmetic formulas for pointed stable curves, extending classical results to an arithmetic setting with applications to lattice volumes.

Findings

Derived explicit formulas for pointed stable curves

Applied formulas to compute volumes of cusp form lattices

Extended classical geometric results to arithmetic context

Abstract

We prove arithmetic Riemann-Roch and Hilbert-Samuel type formulae for pointed stable curves. We give applications to volumes of lattices of integral cusps forms for pointed stable curves of genus 0.