Quadratic Forms in 8 Prime Variables

Abstract We give an asymptotic for the number of prime solutions to $Q(x_{1},\dots , x_{8}) = N$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>8</mml:mn> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> </mml:math> , subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> . 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 $\operatorname {Sp}_{8}(\mathbf{Z}/q\mathbf{Z})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>Sp</mml:mo> <mml:mn>8</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mi>q</mml:mi> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:math> . 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 $\operatorname {SL}_{2}(k)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>SL</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:math> , $k$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> a finite field.