Frobenius’ theta function and Arakelov invariants in genus three

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.

Keywords

Faltings delta-invariant, Kawazumi–Zhang invariant, theta function

2010 Mathematics Subject Classification

Primary 14H15. Secondary 11G50, 14G40, 14H40, 14H42, 14H45.

Full Text (PDF format)

Received 12 November 2019

Accepted 30 January 2020

Published 17 February 2021