Symmetric roots and admissible pairing

TL;DR

This paper offers a geometric interpretation of an adelic invariant related to the self-intersection of the dualising sheaf in hyperelliptic curves, connecting it to symmetric roots and proposing a conjecture about its equivalence to a known invariant.

Contribution

It introduces a new geometric perspective on an adelic invariant using symmetric roots and conjectures its equivalence to Zhang's invariant.

Findings

Provides a geometric interpretation of the invariant

Proposes a conjecture relating two invariants

Connects discriminant modular form with symmetric roots

Abstract

Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant. We provide a simple geometric interpretation for this invariant, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant introduced in this paper coincides with an invariant introduced in a recent paper by S.-W. Zhang.