Character sums of Division polynomials twisted by multiplicative functions

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 obtain nontrivial estimates for multiplicative character sums associated with the division polynomials $\psi_n(P)$ twisted by several multiplicative functions, where $\psi_n(P)$ denotes the $n$-th division polynomial evaluated at $P$.