Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields

Abstract

In this paper we describe an algorithm that quickly computes a maximal a-valued lattice in an F -vector space equipped with a non-degenerate bilinear form, where a is a fractional ideal in a number field F . We then apply this construction to give an algorithm to compute an a-maximal lattice in a quadratic space over any number field F where the prime p = 2 is unramified. We also develop the theory of p-neighbors for a-valued quadratic lattices at an arbitrary prime p of O F (including when p | 2) and prove its close connection to the residual geometry of certain quadrics mod p. Finally we give a well-known application of p-neighboring lattices and exact mass formulas to compute a complete set of representatives for the classes in a given genus of (totally definite) quadratic O F -lattices.