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

2012/08/13 by Jonathan Hanke, Hanke, Jonathan
Computer Science · Mathematics · #11E12 #11E39 #11E41 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1208.2481

openalex publication_date 2012/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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