The Lie Group Structure of Elliptic/Hyperelliptic $\wp$ Functions

TL;DR

This paper explores the Lie group structures underlying elliptic and hyperelliptic Weierstrass $ ext{wp}$ functions up to genus three, revealing specific symmetries and invariants through algebraic and geometric methods.

Contribution

It identifies and characterizes the Lie group structures associated with hyperelliptic $ ext{wp}$ functions for genera one to three, extending understanding of their algebraic symmetries.

Findings

Genus one $ ext{wp}$ functions have SO(2,1) Lie group structure.

Genus two hyperelliptic $ ext{wp}$ functions have SO(3,2) Lie group structure.

Genus three hyperelliptic $ ext{wp}$ functions exhibit SO(9,6) Lie group or subgroup structure.

Abstract

We consider the generalized dual transformation for elliptic/hyperelliptic $\wp$ functions up to genus three. For the genus one case, from the algebraic addition formula, we deduce that the Weierstrass $\wp$ function has the SO(2,1) $\cong$ Sp(2,$\R$)/$\Z_2$ Lie group structure. For the genus two case, by constructing a quadratic invariant form, we find that hyperelliptic $\wp$ functions have the SO(3,2) $\cong$ Sp(4,$\R$)/$\Z_2$ Lie group structure. Making use of quadratic invariant forms reveals that hyperelliptic $\wp$ functions with genus three have the SO(9,6) Lie group and/or it's subgroup structure.