Representations of indefinite quadratic forms. (English) Zbl 0883.11016

Hsia, J. S.; Shao, Y. Y.; Xu, Fei

J. Reine Angew. Math. 494, 129-140 (1998).

Let \(f\) be an integral quadratic form in \(m\geq 3\) variables and defined over an algebraic number field, and \(g\) another in \(n\) variables where \(m\geq n\) and \(n\) is arbitrary. Suppose \(f\) is indefinite. The authors consider the following questions:
(1) How many inequivalent forms in the genus of \(f\) represent \(g\) when one knows that at least one form represents \(g\)?
(2) Is there any algebraic structure to the set of forms which represent \(g\)?
These questions were studied previously when the codimension \(\delta: =m-n\) is greater than or equal to 2, and for the most part when also \(n=1\). The aim of this paper is to give complete answers to these questions.

Reviewer: J.S.Hsia (Columbus)

MSC:

11E12	Quadratic forms over global rings and fields
11D85	Representation problems

Keywords:

representations; indefinite integral quadratic form; spinor genus