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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.

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
© 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).

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