Computing isomorphisms between lattices

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 $2$.