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 $⟨C,C⟩$ of the cuspidal part of the theta series of $Q$. We derive an upper bound on $⟨C,C⟩$ that depends on the smallest positive integer not represented by the dual form $Q^{\ast }$. In addition, we give a non-trivial upper bound on the sum of the integers $n$ excepted by $Q$.

20 pages