Counting integer points on quadrics with arithmetic weights

Abstract

Let F ∈ Z [ x ] F \in \mathbf {Z}[\boldsymbol {x}] be a diagonal, non-singular quadratic form in four variables. Let λ ( n ) \lambda (n) be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer zeros of F F with | x | ⩽ X |\boldsymbol {x}| \leqslant X , weighted by λ ( x 1 ) \lambda (x_1) .