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

We consider diagonal quadratic forms a(1)x(1)(2) +a(2)x(2)(2) + ....+a(l)x(l)(2), 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)D(l-1)/2 > 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.