Bilinear character sums over elliptic curves

Let E be an elliptic curve over a finite field FqFq 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)x(P) to denote the x  -component of a point P=(x(P),y(P))∈EP=(x(P),y(P))∈E.We estimate character sumsWρ,ϑ(ψ,U,V)=∑U∈U∑V∈Vρ(U)ϑ(V)ψ(x(U⊕V)), where UU and VV are arbitrary sets of FqFq-rational points on E, ψ   is a nontrivial additive character of FqFq and ρ(U)ρ(U) and ϑ(V)ϑ(V) are arbitrary bounded complex functions supported on UU and VV, respectively. Our bound of sums Wρ,ϑ(ψ,U,V)Wρ,ϑ(ψ,U,V) is nontrivial whenever#U>q1/2+εand#V>qε for some fixed ε>0ε>0. We also give various applications of this bound.