Representations by Quaternary Quadratic Forms with Coefficients $1$, $2$, $5$ or $10$

TL;DR

This paper derives explicit formulas for counting how many ways positive integers can be represented by specific quaternary quadratic forms with coefficients 1, 2, 5, or 10, using modular forms techniques.

Contribution

It provides explicit formulas for representations by certain quaternary quadratic forms, advancing the understanding of these forms through modular forms methods.

Findings

Explicit formulas for representation counts of positive integers.

Application of modular forms to quadratic form enumeration.

Enhanced understanding of quadratic forms with coefficients 1, 2, 5, 10.

Abstract

We determine explicit formulas for the number of representations of a positive integer $n$ by quaternary quadratic forms with coefficients $1$, $2$, $5$ or $10$. We use a modular forms approach.