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This paper studies two topics concerning on the orthogonal complement of onedimensional subspace with respect to a given quadratic form on a vector spaceover a number field. One is to determine the invariants for the isomorphismclass of such a complement in the sense of Shimura. The other is to investigatean ideal of the base field, which may be viewed as a difference between thegenus of maximal lattices and an integral lattice in the complement. We shalldiscuss about the class number of the genus of maximal lattices as anapplication.

Invariants and discriminant ideals of orthogonal complements in a quadratic space