Special values of canonical Green's functions

TL;DR

This paper derives explicit formulas for the canonical Green's function at Weierstrass points on hyperelliptic Riemann surfaces and relates the energy of these points to spectral invariants, showing it exceeds log 2.

Contribution

It provides new explicit formulas for Green's functions at Weierstrass points and links the energy to spectral invariants, extending known elliptic curve results.

Findings

Explicit formula for Green's function at Weierstrass points

Energy of Weierstrass points exceeds log 2

Connection between energy and spectral invariants

Abstract

We give a precise formula for the value of the canonical Green's function at a pair of Weierstrass points on a hyperelliptic Riemann surface. Further we express the 'energy' of the Weierstrass points in terms of a spectral invariant recently introduced by N. Kawazumi and S. Zhang. It follows that the energy is strictly larger than log 2. Our results generalize known formulas for elliptic curves.