Big prime factors in orders of elliptic curves over finite fields

TL;DR

This paper proves that for large extensions of finite fields, the number of points on an elliptic curve has a large prime factor exceeding a certain growth rate, advancing understanding of elliptic curve group structures.

Contribution

The paper establishes a new lower bound on the size of prime factors in the order of elliptic curves over finite fields for large extension degrees.

Findings

Prime factors of elliptic curve orders grow faster than previously known bounds.

Large prime factors exceed $n ext{exp}(crac{ ext{log} n}{ ext{log} ext{log} n})$ for large $n$.

Provides insights into the distribution of prime factors in elliptic curve groups.

Abstract

Let $E$ be an elliptic curve over the finite field $\mathbb F_q$. We prove that, when $n$ is a sufficiently large positive integer, $\#E(\mathbb F_{q^n})$ has a prime factor exceeding $n\exp(c\log n/\log\log n)$.