Bilinear character sums over elliptic curves

Let E be an elliptic curve over a finite field 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 sums where and are arbitrary sets of -rational points on E, ψ is a nontrivial additive character of and ρ(U) and (V) are arbitrary bounded complex functions supported on and , respectively. Our bound of sums is nontrivial whenever for some fixed ε>0. We also give various applications of this bound.