Computing generators of free modules over orders in group algebras

Abstract

Let E be a number field and G be a finite group. Let A be any O E -order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] ∼ = χ M χ is explicitly computable and each M χ is in fact a matrix ring over a field, this leads to an algorithm that either gives elements α 1 , . . . , α d ∈ X such that X = Aα 1 ⊕ · · · ⊕ Aα d or determines that no such elements exist.Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O L , the ring of algebraic integers of L, and A to be the associated order A(E [G]; O L ) ⊆ E [G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = Q.

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

“…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…”

Computing generators of free modules over orders in group algebras II

“…We briefly recall (with some minor differences and corrections) relevant notation and results from [BJ08,§2]. For further background material we refer the reader to [Rei03].…”

“…(1) If we replace E[G] by a finite-dimensional semisimple E-algebra A (see Remark 4.9), then we define (8) The number of tests for this step can be greatly reduced by using the method described in [BJ08,§7].…”

“…We now give a noncommutative version of [BJ08,Proposition 5.3]. In other words, we solve the problem of determining whether a left Λ-module X is free of finite rank, and if so, whether generators can be computed.…”

“…Let A be any O E -order of full rank in the group algebra E [G] and X a (left) A-lattice, i.e., a (left) A-module that is finitely generated and torsion-free over O E . The main theoretical result of [BJ08] is a necessary and sufficient condition for X to be free of given rank d over A. In order to use this criterion for computational purposes, we had to impose two hypotheses:…”