Invariants and discriminant ideals of orthogonal complements in a quadratic space

TL;DR

This paper investigates the invariants and discriminant ideals associated with orthogonal complements in quadratic spaces over number fields, providing insights into their classification and lattice structures.

Contribution

It introduces a method to determine invariants of orthogonal complements and explores the relationship between discriminant ideals and lattice genus class numbers.

Findings

Invariants for orthogonal complements are explicitly characterized.

Discriminant ideals relate to the genus of maximal lattices.

Class number bounds for lattice genera are discussed.

Abstract

This paper studies two topics concerning on the orthogonal complement of one dimensional subspace with respect to a given quadratic form on a vector space over a number field. One is to determine the invariants for the isomorphism class of such a complement in the sense of Shimura. The other is to investigate an ideal of the base field, which may be viewed as a difference between the genus of maximal lattices and an integral lattice in the complement. We shall discuss about the class number of the genus of maximal lattices as an application.