On the correlations between character sums of division polynomials under shifts

Let $E$ be an elliptic curve over the finite field ${\mathbb{F}}_p$, and $P\in E({\mathbb{F}}_p)$ be an ${\mathbb{F}}_p$-rational point. We study the sums $$S_{\chi ,P}(N,h)=\sum _{n=1}^N\chi ({\psi }_n(P))\chi ({\psi }_{n+h}(P)),$$ where ${\psi }_n(P)$ denotes the $n$-th division polynomial evaluated at $P$, and $\chi$ is a multiplicative character of ${\mathbb{F}}_p^{\ast }$. We estimate $S_{\chi ,P}(N,h)$ on average over $h$ over a rather short interval $h\in [1,H]$. We also obtain a multidimensional generalisation of this result.