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

Article Type

Research Article

Publication Title

Rocky Mountain Journal of Mathematics

Abstract

We consider diagonal quadratic forms a1 x12 + a2 x22 + · · · + aℓxℓ2, where ℓ ≥ 5 is an odd integer and ai ≥ 1 are integers. By using the extended Shimura correspondence, we obtain explicit formulas for the number of representations of | D| n2 by such quadratic forms, where D is either a squarefree integer or a fundamental discriminant such that (−1)(ℓ−1)/2 D > 0. We demonstrate our method with many examples, in particular recovering results of Cooper, Lam and Ye (2013): all their formulas (when ℓ = 5) for n2 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 ai. 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.

First Page

1219

Last Page

1244

DOI

https://10.1216/rmj.2023.53.1219

Publication Date

8-1-2023

Comments

Open Access, Green

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