On the correlations between character sums of division polynomials under shifts

Abstract: 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.