Computing isomorphisms between lattices

TL;DR

This paper develops an algorithm to determine isomorphisms between lattices over orders in semisimple algebras, with applications to Galois module structures in number theory.

Contribution

It generalizes previous algorithms to decide lattice isomorphism under weaker conditions and implements this in Magma for specific group algebras.

Findings

Algorithm successfully determines lattice isomorphisms or non-isomorphisms.

Application to Galois module structures in number fields.

Provides computational tools for algebraic number theory investigations.

Abstract

Let K be a number field, let A be a finite dimensional semisimple K-algebra and let Lambda be an 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 generalise 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 -> Y or determines that X and Y are not isomorphic. The algorithm is implemented in Magma for A=Q[G], Lambda=Z[G] and Lambda-lattices X and Y contained in 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 Q with Galois group isomorphic to Q_8 x C_2, the direct product of the quaternion group of order 8 and the cyclic group of order 2.