Computing generators of free modules over orders in group algebras II

Bley, Werner; Johnston, Henri

doi:10.1090/s0025-5718-2011-02488-9

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…Heuristically, one obtains the best results (in the naive meaning that we get small algebraic numbers with small denominators) if we use an integral normal basis element α 0 . Such an element can be computed by the algorithms developed in [3] and [4]. Henceforth, we assume the rationality conjecture and set…”

Section: Theorem 45 We Assume (I)-(viii) Then the Following Are Eqmentioning

confidence: 99%

Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture (Part II)

Bley¹

2012

Math. Comp.

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Abstract. We continue the study of the Equivariant Tamagawa Number Conjecture for the base change of an elliptic curve begun by the author in 2009. We recall that the methods developed there, apart from very special cases, cannot be applied to verify the l-part of the ETNC if l divides the order of the group. In this note we focus on extensions of l-power degree (l an odd prime) and describe methods for computing numerical evidence for ETNC l . For cyclic l-power extensions we also express the validity of ETNC l in terms of explicit congruences.

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Section: Theorem 45 We Assume (I)-(viii) Then the Following Are Eqmentioning

confidence: 99%

Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture (Part II)

Bley¹

2012

Math. Comp.

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“…Let K/Q be an S 4 -extension with full decomposition group that is wildly ramified. Here we describe how to use the Magma implementation of [5, Algorithm 3.1 (6)] to check whether or not O K is locally free at 2 over A K/Q , or equivalently, whether O K P is free over A K P /Q 2 , where P is the unique prime of K above 2. We used the database [30] to find six number fields, each of which has a completion at 2 equal to one of the six wildly ramified S 4 -extensions of Q 2 listed in the database of p-adic fields [24].…”

Section: Induction For Orders Of a Certain Structurementioning

confidence: 99%

Leopoldt-type theorems for non-abelian extensions of

Ferri

2024

Glasgow Math. J.

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We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions $K/\mathbb{Q}$ with Galois group isomorphic to $A_4$ , $S_4$ , $A_5$ , and dihedral groups of order $2p^n$ for certain prime powers $p^n$ . In particular, when $K/\mathbb{Q}$ is a Galois extension with Galois group $G$ isomorphic to $A_4$ , $S_4$ or $A_5$ , we give necessary and sufficient conditions for the ring of integers $\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra $\mathbb{Q}[G]$ .

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“…In particular, [BE05] applies in the case that A is commutative and semisimple; [BW09] applies to group rings , where G is a finite group, but only decides whether two lattices are both locally free and stably isomorphic; and [DD08, KV10, Pag14] apply to maximal or Eichler orders in quaternion algebras. The series of articles [Ble97, BJ08, BJ11, HJ20] consider progressively more general situations, culminating in a solution to when A is semisimple, but they all involve a very expensive enumeration step, which in many cases renders the algorithm impractical. We refer the reader to the introduction of [HJ20] for a more detailed overview.…”

Section: Introductionmentioning

confidence: 99%

Computation of lattice isomorphisms and the integral matrix similarity problem

Bley

Hofmann

Johnston

2022

Forum of Mathematics, Sigma

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Let K be a number field, let A be a finite-dimensional K-algebra, let $\operatorname {\mathrm {J}}(A)$ denote the Jacobson radical of A and let $\Lambda $ be an $\mathcal {O}_{K}$ -order in A. Suppose that each simple component of the semisimple K-algebra $A/{\operatorname {\mathrm {J}}(A)}$ is isomorphic to a matrix ring over a field. Under this hypothesis on A, we give an algorithm that, given two $\Lambda $ -lattices X and Y, determines whether X and Y are isomorphic and, if so, computes an explicit isomorphism $X \rightarrow Y$ . This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number field K, a positive integer n and two matrices $A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$ , determine whether A and B are similar over $\mathcal {O}_{K}$ , and if so, return a matrix $C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$ such that $B= CAC^{-1}$ . We give explicit examples that show that the implementation of the latter algorithm for $\mathcal {O}_{K}=\mathbb {Z}$ vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.

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Computing generators of free modules over orders in group algebras II

Cited by 5 publications

References 15 publications

Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture (Part II)

Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture (Part II)

Leopoldt-type theorems for non-abelian extensions of

Computation of lattice isomorphisms and the integral matrix similarity problem

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