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. 
Manabu Murata
math.NT