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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Computing isomorphisms between lattices
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by Tommy Hofmann and Henri Johnston HTML | PDF
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