On character sums with distances on the upper half plane over a finite field

For the finite field F_q of q elements (q odd) and a quadratic non-residue (that is, a non-square) α ∈ F_q, we define the distance functionδ (u + v sqrt(α), x + y sqrt(α)) = frac((u - x)² - α (v - y)², v y) on the upper half plane H_q = {x + y sqrt(α) | x ∈ F_q, y ∈ F_q^*} ⊆ F_q2. For two sets E, F ⊂ H_q with # E = E, # F = F and a non-trivial additive character ψ on F_q, we give the following estimate| under(∑, w ∈ E, z ∈ F) ψ (δ (w, z)) | ≤ min {q + sqrt(2 q E), sqrt(3) q^{5 / 4}} sqrt(E F), which (assuming that F ≥ E) is non-trivial if E F / q² → ∞ as q → ∞.