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Almost Prime Orders of Elliptic Curves Over Prime Power Fields

Likun Xie

TL;DR

The paper studies almost-prime phenomena for elliptic curve group orders over prime power fields, motivated by Koblitz's conjectures. It develops CM-field sieve methods to obtain unconditional results for CM curves and leverages GRH-based Chebotarev arguments for non-CM curves, yielding explicit bounds on the number of prime factors in ratios like $\frac{|E(\mathbb{F}_{p^\ell})|}{|E(\mathbb{F}_p)|}$. For the CM curve $E: y^2 = x^3 - x$, it demonstrates that infinitely many primes $p$ yield $|E(\mathbb{F}_{p^2})|/32$ as a square-free almost-prime, and more broadly that $\Omega(|E(\mathbb{F}_{p^\ell})|/|E(\mathbb{F}_p)|)$ is bounded by a function of $\ell$, with explicit counts $\gg x/(\log x)^2$. In the non-CM setting, assuming GRH, similar bounds hold for odd primes $\ell$, with the density controlled by division-field Galois representations. Collectively, the work advances Koblitz-type primality questions for elliptic curves using a blend of Bombieri–Vinogradov in CM fields, Chebotarev density, and vector/linear sieve techniques, and it highlights structural consequences for the group architectures $E(\mathbb{F}_p)$ and $E(\mathbb{F}_{p^2})$.

Abstract

In 1988, Koblitz conjectured the infinitude of primes p for which |E(F_p)| is prime for elliptic curves E over Q, drawing an analogy with the twin prime conjecture. He also proposed studying the primality of |E(F_{p^l})| / |E(F_p)|, in parallel with the primality of (p^l - 1)/(p - 1). Motivated by these problems and earlier work on |E(F_p)|, we study the infinitude of primes p such that |E(F_{p^l})| / |E(F_p)| has a bounded number of prime factors for primes l >= 2, considering both CM and non-CM elliptic curves over Q. In the CM case, we focus on the curve y^2 = x^3 - x to address gaps in the literature and present a more concrete argument. The result is unconditional and applies Huxley's large sieve inequality for the associated CM field. In the non-CM case, analogous results follow under GRH via the effective Chebotarev density theorem. For the CM curve y^2 = x^3 - x, we further apply a vector sieve to combine the almost prime properties of |E(F_p)| and |E(F_{p^2})| / |E(F_p)|, establishing a lower bound for the number of primes p <= x for which |E(F_{p^2})| / 32 is a square-free almost prime. We also study cyclic subgroups of finite index in E(F_p) and E(F_{p^2}) for CM curves.

Almost Prime Orders of Elliptic Curves Over Prime Power Fields

TL;DR

The paper studies almost-prime phenomena for elliptic curve group orders over prime power fields, motivated by Koblitz's conjectures. It develops CM-field sieve methods to obtain unconditional results for CM curves and leverages GRH-based Chebotarev arguments for non-CM curves, yielding explicit bounds on the number of prime factors in ratios like . For the CM curve , it demonstrates that infinitely many primes yield as a square-free almost-prime, and more broadly that is bounded by a function of , with explicit counts . In the non-CM setting, assuming GRH, similar bounds hold for odd primes , with the density controlled by division-field Galois representations. Collectively, the work advances Koblitz-type primality questions for elliptic curves using a blend of Bombieri–Vinogradov in CM fields, Chebotarev density, and vector/linear sieve techniques, and it highlights structural consequences for the group architectures and .

Abstract

In 1988, Koblitz conjectured the infinitude of primes p for which |E(F_p)| is prime for elliptic curves E over Q, drawing an analogy with the twin prime conjecture. He also proposed studying the primality of |E(F_{p^l})| / |E(F_p)|, in parallel with the primality of (p^l - 1)/(p - 1). Motivated by these problems and earlier work on |E(F_p)|, we study the infinitude of primes p such that |E(F_{p^l})| / |E(F_p)| has a bounded number of prime factors for primes l >= 2, considering both CM and non-CM elliptic curves over Q. In the CM case, we focus on the curve y^2 = x^3 - x to address gaps in the literature and present a more concrete argument. The result is unconditional and applies Huxley's large sieve inequality for the associated CM field. In the non-CM case, analogous results follow under GRH via the effective Chebotarev density theorem. For the CM curve y^2 = x^3 - x, we further apply a vector sieve to combine the almost prime properties of |E(F_p)| and |E(F_{p^2})| / |E(F_p)|, establishing a lower bound for the number of primes p <= x for which |E(F_{p^2})| / 32 is a square-free almost prime. We also study cyclic subgroups of finite index in E(F_p) and E(F_{p^2}) for CM curves.

Paper Structure

This paper contains 18 sections, 28 theorems, 360 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Main Results
  4. Comparisons in the CM and Non-CM Cases
  5. Notation
  6. Sieving Tools
  7. The Bombieri--Vinogradov Theorem for Algebraic Number Fields
  8. The Effective Version of the Chebotarev Density Theorem
  9. Vector sieve of linear sieves
  10. Orders of $E(\mathbb{F}_p)$ and $E(\mathbb{F}_{p^2})$ for CM elliptic curves
  11. Sieve Set-up and Weighted Terms
  12. Estimation of $W(x)$
  13. Estimation of $r(d_1,d_2)$
  14. Estimation of the sieving functions
  15. Group Structures of $E(\mathbb{F}_p)$ and $E(\mathbb{F}_{p^2})$
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $E /\mathbb{Q}$ be a CM elliptic curve such that all curves $\mathbb{Q}$-isogenous to $E$ have trivial $\mathbb{Q}$-torsion. Let $N_E$ be the conductor of $E$. Then there exists a constant $C(E) > 0$ such that as $x \to \infty$,

Theorems & Definitions (51)

  • Conjecture 1.1
  • Theorem 1.2: cojocaru
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • ...and 41 more