Bilinear character sums over elliptic curves

Let E be an elliptic curve over a finite field F_q of q elements given by an affine Weierstraß equation. Let ⊕ denote the group operation in the Abelian group of points on E. We also use x (P) to denote the x-component of a point P = (x (P), y (P)) ∈ E. We estimate character sumsW_{ρ, θ{symbol}} (ψ, U, V) = under(∑, U ∈ U) under(∑, V ∈ V) ρ (U) θ{symbol} (V) ψ (x (U ⊕ V)), where U and V are arbitrary sets of F_q-rational points on E, ψ is a nontrivial additive character of F_q and ρ (U) and θ{symbol} (V) are arbitrary bounded complex functions supported on U and V, respectively. Our bound of sums W_{ρ, θ{symbol}} (ψ, U, V) is nontrivial whenever# U > q^{1 / 2 + ε} and # V > q^ε for some fixed ε > 0. We also give various applications of this bound.