Frobenius' theta function and Arakelov invariants in genus three

TL;DR

This paper derives explicit formulas for key Arakelov invariants of genus three Riemann surfaces using Frobenius' theta function, connecting classical theta functions with modern height pairing results.

Contribution

It provides the first explicit formulas for Kawazumi-Zhang and Faltings invariants in genus three, utilizing Frobenius' theta function and relating to recent height pairing formulas.

Findings

Explicit formulas for Kawazumi-Zhang and Faltings delta-invariants.

Connection between Frobenius' theta function and Arakelov invariants.

Link to height pairing of Ceresa cycles in genus three.

Abstract

We give explicit formulas for the Kawazumi-Zhang invariant and Faltings delta-invariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral involving the standard Riemann theta function, and another involving a theta function particular to genus three that was discovered by Frobenius. We review part of Frobenius' work on his theta function and connect our results with a formula due to Bloch, Hain and Bost describing the archimedean height pairing of Ceresa cycles in genus three.