Quadratic forms in 8 prime variables

Journal article

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

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Copyright holder:: Ben Green
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