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 $\mathbb{F}_p^{*}$. 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.