Quaternary even positive definite quadratic forms of prime discriminant

Abstract

In this note we study quaternary even positive definite quadratic forms of prime discriminant. In § 1 we classify quaternary even positive definite quadratic forms of prime discriminant p = 1 mod 4 (called simply nice quaternary lattices in this note) which represent two. We note that the class number of such forms is closely related to the dimension of the space of certain automorphic forms. (Remark 4 in the text). By using the classification in § 1 and the theory of integral representations of cyclic groups we show that the orthogonal group of a nice quaternary lattice is generated by ±1 and symmetries (of the lattice). In § 3, we calculate the class number of nice quaternary lattices. Notations and terminologies will generally be those of O’Meara [5]. Any exceptions to this convention will be stated explicitly.