Quaternary quadratic forms with prime discriminant

TL;DR

This paper establishes explicit lower bounds for the number of representations of positive integers by positive-definite quaternary quadratic forms with prime discriminant, linking it to bounds on the Petersson norm of associated theta series.

Contribution

It provides a novel explicit lower bound on representations and connects it to bounds on the Petersson norm, depending on the smallest non-represented integer by the dual form.

Findings

Explicit lower bounds on representation counts for prime discriminant forms.

Upper bounds on Petersson norms related to non-represented integers.

Non-trivial bounds on the sum of integers not represented by the form.

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 $Q$.