We give an asymptotic for the number of prime solutions to $Q(x_{1},\dots , x_{8}) = N$ , subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$ .
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 $\operatorname {Sp}_{8}(\mathbf{Z}/q\mathbf{Z})$ . 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 $\operatorname {SL}_{2}(k)$ , $k$ a finite field.
