Reduction theory for stably graded Lie algebras

Thorne, Jack A.

doi:10.2140/ant.2026.20.195

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Reduction theory for stably graded Lie algebras

Jack A. Thorne

Vol. 20 (2026), No. 1, 195–208

DOI: 10.2140/ant.2026.20.195

Abstract

We define a reduction covariant for the representations à 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.

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Keywords

reduction theory, arithmetic groups

Mathematical Subject Classification

Primary: 11F06, 20G20

Milestones

Received: 3 April 2024

Revised: 13 December 2024

Accepted: 20 January 2025

Published: 28 November 2025

Authors

Jack A. Thorne
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge CB3 0WB United Kingdom

Vol. 20, No. 1, 2026