FINITENESS THEOREMS FOR POSITIVE DEFINITE n-REGULAR QUADRATIC FORMS

An integral quadratic form f of m variables is said to be n-regular if f globally represents all quadratic forms of n variables that are represented by the genus of f . For any n ≥ 2, it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of n+3 variables that are n-regular. We also investigate similar finiteness results for almost n-regular and spinor n-regular quadratic forms. It is shown that for any n ≥ 2, there are only finitely many equivalence classes of primitive positive definite spinor or almost n-regular quadratic forms of n+ 2 variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).