Invariants and discriminant ideals of orthogonal complements in a quadratic space

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.