On the Sum-Product Problem on Elliptic Curves

TL;DR

This paper investigates sum-product type problems on elliptic curves over finite fields, demonstrating that certain combined sets derived from elliptic curve points are necessarily large, extending sum-product phenomena to elliptic curve contexts.

Contribution

The paper establishes lower bounds on the size of sum and product sets of elliptic curve points, extending sum-product results to the setting of elliptic curves over finite fields.

Findings

Either the sum set or the product set is large for subsets of the elliptic curve points.

The results connect sum-product phenomena with elliptic curve arithmetic.

Provides new insights into algebraic structures over finite fields.

Abstract

Let $\E$ be an ordinary elliptic curve over a finite field $\F_{q}$ of $q$ elements and $x(Q)$ denote the $x$-coordinate of a point $Q = (x(Q),y(Q))$ on $\E$. Given an $\F_q$-rational point $P$ of order $T$, we show that for any subsets $\cA, \cB$ of the unit group of the residue ring modulo $T$, at least one of the sets $$ \{x(aP) + x(bP) : a \in \cA, b \in \cB\} \quad\text{and}\quad \{x(abP) : a \in \cA, b \in \cB\} $$ is large. This question is motivated by a series of recent results on the sum-product problem over finite fields and other algebraic structures.