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The probability of isomorphic group structures of isogenous elliptic curves over finite fields

John Cullinan, Nathan Kaplan

TL;DR

The paper studies the density of primes p for which two fixed ℓ-isogenous elliptic curves E and E' over ℚ have isomorphic groups $E(\mathbf{F}_p)$ and $E'(\mathbf{F}_p)$. It develops a Chebotarev-density framework using ℓ-adic and mod ℓ^n representations, defining quantities $d_{\ell^m}$ and $d'_{\ell^m}$ from the ℓ-adic image data, and proves a universal density formula $P(E,E') = 1 - \sum_{m=1}^\infty \left( \frac{d_{\ell^m}}{|G(\ell^m)|} + \frac{d'_{\ell^m}}{|G'(\ell^m)|} \right)$ that yields the limiting proportion of primes where the groups are nonisomorphic. For non-CM curves the $d_{\ell^m}$ stabilize to $1-1/\ell$ and the tail simplifies into a geometric sum; in CM cases, the density is 1 for ℓ>3 with a handful of exceptional smaller ℓ values. The paper provides exact computations and extensive examples, including CM and non-CM instances, as well as the effect of quadratic twists on the density. The results give a precise, broadly applicable method to quantify how often isomorphic reductions occur across primes for isogenous elliptic curves, with practical computations demonstrated via explicit data and examples.

Abstract

Let l be a prime number and let E and E' be l-isogenous elliptic curves defined over Q. In this paper we determine the proportion of primes p for which E(F_p) is isomorphic to E'(F_p). Our techniques are based on those developed in \cite{ck} and \cite{rnt}.

The probability of isomorphic group structures of isogenous elliptic curves over finite fields

TL;DR

The paper studies the density of primes p for which two fixed ℓ-isogenous elliptic curves E and E' over ℚ have isomorphic groups and . It develops a Chebotarev-density framework using ℓ-adic and mod ℓ^n representations, defining quantities and from the ℓ-adic image data, and proves a universal density formula that yields the limiting proportion of primes where the groups are nonisomorphic. For non-CM curves the stabilize to and the tail simplifies into a geometric sum; in CM cases, the density is 1 for ℓ>3 with a handful of exceptional smaller ℓ values. The paper provides exact computations and extensive examples, including CM and non-CM instances, as well as the effect of quadratic twists on the density. The results give a precise, broadly applicable method to quantify how often isomorphic reductions occur across primes for isogenous elliptic curves, with practical computations demonstrated via explicit data and examples.

Abstract

Let l be a prime number and let E and E' be l-isogenous elliptic curves defined over Q. In this paper we determine the proportion of primes p for which E(F_p) is isomorphic to E'(F_p). Our techniques are based on those developed in \cite{ck} and \cite{rnt}.
Paper Structure (6 sections, 8 theorems, 23 equations, 1 table)

This paper contains 6 sections, 8 theorems, 23 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Theorem
  4. Main Result
  5. Examples
  6. Remarks on CM Elliptic Curves

Key Result

Proposition 1.2

With notation as above, suppose the elliptic curves $E$ and $E'$ do not have CM and that the maximum of the levels of their $\ell$-adic representations is $\ell^M$. Then we have $d_{\ell^m}, d'_{\ell^m} \in \lbrace 0,1 - 1/\ell \rbrace$ for all $m \geq 1$ and $d_{\ell^m} = d'_{\ell^m} = 1-1/\ell$ f

Theorems & Definitions (23)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 13 more