Counting integer points on quadrics with arithmetic weights

Abstract

Let $F\in \mathbf{Z}[\mathbf{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda (n)$ 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 $F$ with $|\mathbf{x}|\le X$, weighted by $\lambda (x_1)$.

36 pages. The article has been substantially revised after incorporating the referee's suggestions