# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Computing isomorphisms between latticesHTML articles powered by AMS MathViewer

by Tommy Hofmann and Henri Johnston
Math. Comp. 89 (2020), 2931-2963 Request permission

## Abstract:

Let $K$ be a number field, let $A$ be a finite-dimensional semisimple $K$-algebra, and let $\Lambda$ be an $\mathcal {O}_{K}$-order in $A$. It was shown in previous work that, under certain hypotheses on $A$, there exists an algorithm that for a given (left) $\Lambda$-lattice $X$ either computes a free basis of $X$ over $\Lambda$ or shows that $X$ is not free over $\Lambda$. In the present article, we generalize this by showing that, under weaker hypotheses on $A$, there exists an algorithm that for two given $\Lambda$-lattices $X$ and $Y$ either computes an isomorphism $X \rightarrow Y$ or determines that $X$ and $Y$ are not isomorphic. The algorithm is implemented in Magma for $A=\mathbb {Q}[G]$, $\Lambda =\mathbb {Z}[G]$, and $\Lambda$-lattices $X$ and $Y$ contained in $\mathbb {Q}[G]$, where $G$ is a finite group satisfying certain hypotheses. This is used to investigate the Galois module structure of rings of integers and ambiguous ideals of tamely ramified Galois extensions of $\mathbb {Q}$ with Galois group isomorphic to $Q_{8} \times C_{2}$, the direct product of the quaternion group of order $8$ and the cyclic group of order $2$.
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Additional Information
• Tommy Hofmann
• Affiliation: Fachbereich Mathematik, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany
• MR Author ID: 1074375
• Email: thofmann@mathematik.uni-kl.de
• Henri Johnston
• Affiliation: Department of Mathematics, University of Exeter, Exeter, EX4 4QF United Kingdom
• MR Author ID: 776746
• ORCID: 0000-0001-5764-0840
• Email: H.Johnston@exeter.ac.uk
• Received by editor(s): March 2, 2019
• Received by editor(s) in revised form: January 13, 2020
• Published electronically: June 1, 2020
• Additional Notes: The first author was supported by Project II.2 of SFB-TRR 195 “Symbolic Tools in Mathematics and their Application” of the German Research Foundation (DFG)
The second author was supported by EPSRC First Grant EP/N005716/1 “Equivariant Conjectures in Arithmetic”.
• © Copyright 2020 American Mathematical Society
• Journal: Math. Comp. 89 (2020), 2931-2963
• MSC (2010): Primary 11R33, 11Y40, 16Z05
• DOI: https://doi.org/10.1090/mcom/3543
• MathSciNet review: 4136552