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

TL;DR

This paper introduces algorithms for efficiently computing maximal lattices in bilinear and quadratic spaces over number fields, with applications to class enumeration and residual geometry analysis.

Contribution

It presents new algorithms for computing maximal a-valued lattices and develops the theory of p-neighbors for quadratic lattices over number fields.

Findings

Algorithm for quickly computing maximal a-valued lattices.

Development of p-neighbor theory for quadratic lattices at arbitrary primes.

Application to class enumeration using mass formulas.

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 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.