Representations of squares by certain diagonal quadratic forms in odd number of variables

TL;DR

This paper develops explicit formulas for representing numbers by certain diagonal quadratic forms with an odd number of variables, using extended Shimura correspondence, generalizing previous results and deriving new identities and congruences.

Contribution

The paper introduces a general method using extended Shimura correspondence to obtain formulas for representations by diagonal quadratic forms with arbitrary coefficients, extending prior work.

Findings

Explicit formulas for representation numbers involving |D|n^2.

Recovery of known formulas for specific cases when =5.

Derivation of identities and congruences related to Fourier coefficients and divisor functions.

Abstract

In this paper, we consider the following diagonal quadratic forms \begin{equation*} a_1x_1^2 + a_2x_2^2 + \cdots + a_{\ell}x_{\ell}^2, \end{equation*} where $\ell\ge 5$ is an odd integer and $a_i\ge 1$ are integers. By using the extended Shimura correspondence, we obtain explicit formulas for the number of representations of $|D|n^2$ by the above type of quadratic forms, where $D$ is either a square-free integer or a fundamental discriminant such that $(-1)^{(\ell-1)/2}D > 0$. We demonstrate our method with many examples, in particular, we obtain all the formulas (when $\ell =5$) obtained in the work of Cooper-Lam-Ye (Acta. Arith. 2013) and all the representation formulas for $n^2$ obtained by them in (Integers, 2013) when $n$ is even. The works of Cooper et. al make use of certain theta function identities combined with a method of Hurwitz to derive these formulas. It is to be noted that our method works in general with arbitrary coefficients $a_i$. As a consequence to some of our formulas, we obtain certain identities among the representation numbers and also some congruences involving Fourier coefficients of certain newforms of weights $6, 8$ and the divisor functions.