On the correlations between character sums of division polynomials under shifts

Let E be an elliptic curve over the finite field Fₚ, and P ∈ E(Fₚ) be an Fₚ-rational point. We study the sums \[ Sχ,P(N,h) = ∑ₙ₌₁^N χ(ψ_n(P)) χ(ψₙ₊ₕ(P)), \] where ψₙ(P) denotes the n-th division polynomial evaluated at P, and $χ$ is a multiplicative character of Fₚ*. We estimate Sχ,P(N,h) on average over h over a rather short interval h ∈ [1, H]. We also obtain a multidimensional generalisation of this result.