REPRESENTATIONS OF SQUARES BY CERTAIN DIAGONAL QUADRATIC FORMS IN AN ODD NUMBER OF VARIABLES

Abstract

We consider diagonal quadratic forms

$$a_1{x}_1^2+a_2{x}_2^2+\cdots +a_{\ell }{x}_{\ell }^2,$$

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 $\left|D\right|n^2$ by such quadratic forms, where $D$ is either a squarefree integer or a fundamental discriminant such that . We demonstrate our method with many examples, in particular recovering results of Cooper, Lam and Ye (2013): all their formulas (when $\ell =5$) for $n^2$ for quinary quadratic forms and all the representation formulas for septenary quadratic forms when $n$ is even. (Those formulas were originally derived by combining certain theta function identities with a method of Hurwitz.) Our method works with arbitrary coefficients $a_i$. As a consequence of some of our formulas, we obtain identities among the representation numbers and also congruences involving the Fourier coefficients of certain newforms of weights $6$ and $8$ and divisor functions.