Automorphisms of Torsion Quadratic Forms

2012-08-28T17:26:51Z

Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = \langle e_i, e_j \rangle$, where $\langle x,y \rangle = q(x+y) - q(x) - q(y)$ is the associated bilinear form. Let $L^*$ be the dual lattice to $L$, and $G_L = L/L^*$, and assume $|G_L|$ is odd. $q$ then descends to a $\mathbb{Q}/\mathbb{Z}$-valued quadratic form $q_G$ on $G_L$. Let $O(G_L)$ be the group of automorphisms of $G_L$ that preserve $q_G$.

Is there an algorithm to compute $O(G_L)$ given $K_{ij}$? What possible groups $O(G_L)$ can arise in this way? One might obtain some information by studying the natural map $f: O(L) \rightarrow O(G_L)$ (where $O(L)$ is the group of automorphisms of $L$ preserving $q$), but $f$ is not necessarily surjective (although its cokernel may be easily computable). However, $f$ might also have a large kernel, and $O(L)$ itself may be difficult to compute. Is there an easier way to compute $O(G_L)$?

Since these questions are somewhat open ended, any useful references would also be appreciated. Thank you.

(edit: corrected mistake above, thanks to B. Conrad for pointing it out)

Automorphisms of Torsion Quadratic Forms - MathOverflow

Automorphisms of Torsion Quadratic Forms