Theta series of ternary and quaternary quadratic forms

Abstract

In the theory of integral quadratic forms or in the allied subject of modular forms very little seems to be known about the linear independence of theta series associated to positive definite forms. For a system of inequivalent positive binary quadratic forms with a fixed fundamental discriminant, Hecke [4-1 showed that their theta series are independent. However, for positive forms in three or more variables, even in a fixed genus, there does not appear to be any non-trivial general result of a similar nature in that certain explicitly described forms in the genus yield linearly independent theta series. We consider here even positive definite ternary quadratic forms of discriminant 2p and quaternary quadratic forms of discriminant p where p is a prime congruent to 1 mod4. For a fixed such p these forms belong to a single genus G'.=G(3,2p), resp. G(4,p). Let G' denote the subset consisting of those forms in G which represent 2. In [6] Kitaoka observed that the number of form-classes in G'(4, p)