The Growth of Class Numbers of Quadratic Forms

Abstract

The Hasse-Minkowski Theorem reduces the isometry question for quadratic spaces over global fields to the corresponding problem over local fields. But the analogous result does not hold in the integral situation. Thus if F is an algebraic number field and o is the ring of integers of F, then two o-lattices on a quadratic F-space V may be in the same genus (that is, locally isometric at all primes p) while not in the same isometry class. The determination or estimation of the class number (the number of classes in the genus) of an arbitrary o-lattice on a quadratic F-space in terms of a computable set of invariants remains a major open problem, although some progress has been made in special cases. For example, it has been shown (see