Reduction theory for stably graded Lie algebras

TL;DR

This paper introduces a reduction theory for stably graded Lie algebras, providing tools for effective coefficient reduction in representations related to 2-descent on odd hyperelliptic curves.

Contribution

It develops a reduction covariant for Vinberg-type representations of stably graded Lie algebras and adapts the LLL algorithm for the odd split special orthogonal group.

Findings

Defined a reduction covariant for Vinberg representations

Extended LLL algorithm to the odd split special orthogonal group

Enabled effective coefficient reduction in hyperelliptic curve computations

Abstract

We define a reduction covariant for the representations a la Vinberg associated to stably graded Lie algebras. We then give an analogue of the LLL algorithm for the odd split special orthogonal group and show how this can be combined with our theory to effectively reduce the coefficients of vectors in a representation connected to 2-descent for odd hyperelliptic curves.