On the correlations between character sums of division polynomials under shifts

TL;DR

This paper investigates correlations between character sums of division polynomials on elliptic curves over finite fields, providing average estimates over shifts and a multidimensional extension.

Contribution

It introduces new average bounds for character sums of division polynomials on elliptic curves, including a multidimensional generalization.

Findings

Derived bounds for sums over short intervals of shifts

Extended results to multidimensional shift scenarios

Provided insights into the distribution of division polynomial character sums

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.