## Quaternary quadratic forms with prime discriminant

### Volume 209 / 2023

Acta Arithmetica 209 (2023), 191-217
MSC: Primary 11E20; Secondary 11F27, 11F30, 11E12.
DOI: 10.4064/aa220601-14-7
Published online: 18 October 2023

#### 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 $Q$.