Finiteness theorems for positive definite 𝑛-regular quadratic forms

Abstract

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