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On the Density of Coprime Reductions of Elliptic Curves

Asimina S. Hamakiotes, Sung Min Lee, Jacob Mayle, Tian Wang

Abstract

Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be viewed as an elliptic curve analogue of the classical question concerning the density of coprime integer pairs. Motivated by Zywina's refinement of the Koblitz conjecture, we formulate a conjecture for the density of such primes. We prove that the series defining this constant converges and admits an almost Euler product expansion. In the case of Serre pairs, we give a closed formula for the constant and use it to prove a moments result describing the distribution of these constants as $(E_1, E_2)$ varies.

On the Density of Coprime Reductions of Elliptic Curves

Abstract

Given non-CM elliptic curves and over , we study the natural density of primes of good reduction for which the orders of the groups and are coprime. This problem may be viewed as an elliptic curve analogue of the classical question concerning the density of coprime integer pairs. Motivated by Zywina's refinement of the Koblitz conjecture, we formulate a conjecture for the density of such primes. We prove that the series defining this constant converges and admits an almost Euler product expansion. In the case of Serre pairs, we give a closed formula for the constant and use it to prove a moments result describing the distribution of these constants as varies.

Paper Structure

This paper contains 14 sections, 29 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fiber Products of Groups
  4. Galois Representations of Elliptic Curves
  5. Serre Curves
  6. Galois Representations for Products of Elliptic Curves
  7. Serre Pairs
  8. A Heuristic Model
  9. Counting Matrices
  10. Coprimality Constant for Serre Pairs
  11. Average Constants and Moments
  12. Numerical Examples
  13. An Example with Only Finitely Many Primes of Coprime Reduction
  14. Serre Pair Examples

Key Result

Theorem 3

For any positive integer $t$, as $T \to \infty$, where $\mathcal{E}(T)$ is defined in E:ET and the constant $C^{\mathrm{coprime}}$ is defined by

Theorems & Definitions (59)

  • Conjecture 2: Coprimality Conjecture
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Lemma 6
  • Lemma 7
  • proof
  • Theorem 8: Serre, MR0387283
  • Proposition 9
  • proof
  • ...and 49 more