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.
Categories: math.NT