Quadratic forms in 8 prime variables

TL;DR

This paper establishes an asymptotic count for prime solutions to quadratic forms in eight variables, combining the circle method with advanced representation theory and harmonic analysis techniques.

Contribution

It introduces a novel approach by interpreting exponential sums as matrix coefficients of the Weil representation, enabling bounds via an amplification method.

Findings

Derived an asymptotic formula for prime solutions to quadratic forms in 8 variables.

Developed a new method linking exponential sums to Weil representation matrix coefficients.

Bounded averages of matrix coefficients using harmonic analysis and representation theory.

Abstract

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.