A quadratic Vinogradov mean value theorem in finite fields

Let p be a prime, let s ≥ 3 be a natural number and let A ⊆ Fₚ be a non-empty set satisfying |A| p1/2. Denoting Jₛ(A) to be the number of solutions to the system of equations ∑ᵢ₌₁ˢ (xᵢ - xᵢ₊ₛ) = ∑ᵢ₌₁ˢ (xᵢ² - xᵢ₊ₛ²) = 0, with x₁, , x₂ₛ ∈ A, our main result implies that Jₛ(A) |A|2s - 2 - 1/9. 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 × A and a special family of modular hyperbolae.