Quadratic Forms in 8 Prime Variables

Abstract

We give an asymptotic for the number of prime solutions to $Q(x_{1},\dots , x_{8}) = N$ Q ( x 1 , … , x 8 ) = N , subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$ Q . The argument initially proceeds via the circle method, but this does not suffice by itself. To obtain a nontrivial bound on certain averages of exponential sums, we interpret these sums as matrix coefficients for the Weil representation of the symplectic group $\operatorname {Sp}_{8}(\mathbf{Z}/q\mathbf{Z})$ Sp 8 ( Z / q Z ) . Averages of such matrix coefficients are then bounded using an amplification argument and a convergence result for convolutions of measures, which reduces matters to understanding the action of certain 12-dimensional subgroups in the Weil representation. Sufficient understanding can be gained by using the basic represention theory of $\operatorname {SL}_{2}(k)$ SL 2 ( k ) , $k$ k a finite field.