A Quadratic Vinogradov Mean Value Theorem in Finite Fields

Let $p$ be a prime, let $s\ge 3$ be a natural number and let $A\subseteq {\mathbb{F}}_p$ be a non-empty set satisfying $\mathrm{\mid }A\mathrm{\mid }\ll p^{1\mathrm{/}2}$. Denoting $J_s(A)$ to be the number of solutions to the system of equations $$\sum _{i=1}^s(x_i-x_{i+s})=\sum _{i=1}^s(x_i^2-x_{i+s}^2)=0,$$ with $x_1,\dots ,x_{2s}\in A$, our main result implies that $$J_s(A)\ll \mathrm{\mid }A{\mathrm{\mid }}^{2s-2-1\mathrm{/}9}\mathrm{.}$$ This can be seen as a finite field analogue of the quadratic Vinogradov mean value theorem. Our techniques involve a variety of combinatorial geometric estimates, including studying incidences between cartesian products $A\times A$ and a special family of modular hyperbolae.