# The Genus Two Free Boson in Arakelov Geometry

## TL;DR

This paper computes the genus two free boson partition function using Arakelov geometry, expressing it as a product of modular forms and analyzing its modular properties and degeneration behavior.

## Contribution

It provides a novel computation of the genus two partition function in Arakelov geometry, extending known genus one results to higher genus.

## Key findings

- Partition function expressed as a product of modular forms
- Confirmed expected obstruction to holomorphic factorization
- Analyzed behavior under degeneration of Riemann surfaces

## Abstract

Using Arakelov geometry, we compute the partition function of the noncompact free boson at genus two. We begin by compiling a list of modular invariants which appear in the Arakelov theory of Riemann surfaces. Using these quantities, we express the genus two partition function as a product of modular forms, as in the well-known genus one case. We check that our result has the expected obstruction to holomorphic factorization and behavior under degeneration.