Period Constraints on Hyperelliptic Branch Points

TL;DR

This paper explores the constraints on the periods of hyperelliptic curves using information theoretic analysis, revealing limitations on the possible period matrices and their relationship to branch points.

Contribution

It constructs a canonical homology basis for hyperelliptic curves and identifies specific constraints on their period matrices, defining an open set in the Siegel upper half space that hyperelliptic periods cannot occupy.

Findings

Certain period matrices are impossible for hyperelliptic curves.

A canonical homology basis clarifies the period-branch point relationship.

Defines an open set in the Siegel upper half space excluded by hyperelliptic periods.

Abstract

Information Theoretic analysis of the periods of a hyperelliptic curve provides more information about the well--known but abstract relationship between the branch points and the periods. Here one constructs a canonical homology basis for a hyperelliptic curve that shows that its periods must satisfy certain constraints and defines an open set in the Siegel upper half space that cannot contain any period matrices of hyperelliptic curves.