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

For the finite field IFq of q elements (q odd) and a quadratic nonresidue α ∈ IFq, we define the distance function δ ( u+ v √ α, x+ y √ α ) = (u− x)2 − α(v − y)2 vy on the upper half plane Hq = {x + y √ α | x ∈ IFq, y ∈ IFq} ⊆ IFq2 . For two sets E ,F ⊂ Hq with #E = E, #F = F and a non-trivial additive character ψ on IFq, we give the following estimate ∣∣∣∣∣ ∑ w∈E,z∈F ψ(δ(w, z)) ∣∣∣∣∣ ≤ min {√ 3q, q + √ 2qE }√ EF, which is non-trivial if EF/q2 →∞ as q →∞. 2000 Mathematics Subject Classification: 11T23, 52C10