Character sums of Division polynomials twisted by multiplicative functions

TL;DR

This paper provides new estimates for character sums involving division polynomials of elliptic curves over finite fields, twisted by multiplicative functions, advancing understanding of their arithmetic properties.

Contribution

It introduces nontrivial bounds for multiplicative character sums of division polynomials evaluated at rational points on elliptic curves, twisted by various multiplicative functions.

Findings

Established nontrivial bounds for character sums involving division polynomials.

Extended analysis to sums twisted by multiple multiplicative functions.

Enhanced understanding of the arithmetic behavior of elliptic curve division polynomials.

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 $P$.