Quaternary quadratic forms with prime discriminant

Abstract

Let \(Q\) be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer \(n\) by \(Q\). This problem is connected with deriving an upper bound on the Petersson norm \(\langle C, C \rangle\) of the cuspidal part of the theta series of \(Q\). We derive an upper bound on \(\langle C, C \rangle\) that depends on the smallest positive integer not represented by the dual form \(Q^{*}\). In addition, we give a non-trivial upper bound on the sum of the integers \(n\) excepted by \(Q\).