Quadratic Forms in 8 Prime Variables

We give an asymptotic for the number of prime solutions to Q(x₁, , x₈) = 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 Sp₈(Z/qZ) 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 SL₂(k) SL 2 ( k ) , k k a finite field.