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The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II

John Cullinan, Shanna Dobson, Linda Frey, Asimina Hamakiotes, Roberto Hernandez, Nathan Kaplan, Jorge Mello, Gabrielle Scullard

Abstract

Let $E$ and $E'$ be 2-isogenous elliptic curves over $\Q$. Following \cite{ck}, we call a good prime $p$ \emph{anomalous} if $E(\F_p) \simeq E'(\F_p)$ but $E(\F_{p^2}) \not \simeq E'(\F_{p^2})$. Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II

Abstract

Let and be 2-isogenous elliptic curves over . Following \cite{ck}, we call a good prime \emph{anomalous} if but . Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.
Paper Structure (23 sections, 30 theorems, 83 equations, 1 figure)

This paper contains 23 sections, 30 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Background and First Definitions
  4. Main Results
  5. Notation
  6. Acknowledgments
  7. Background
  8. Prior Work
  9. Definitions, Terminology, and Setup
  10. Frobenius Properties of Anomalous Primes
  11. Galois Theory of Torsion Fields
  12. Prime Decompositions
  13. Elliptic Curves without CM
  14. Extremal Values
  15. Elliptic Curves with CM
  16. ...and 8 more sections

Key Result

Proposition 1.4

A good prime $p$ for $E$ and $E'$ has defect $(m+1,m)$ if and only if with similar congruences for primes of defect $(m,m+1)$.

Figures (1)

  • Figure 1: Isogeny-Torsion Graphs of CM curves with a point of order 2

Theorems & Definitions (71)

  • Definition 1.3
  • Proposition 1.4
  • Remark 1.5
  • Definition 1.6
  • Proposition 1.8
  • Theorem 1.9
  • Remark 1.10
  • Theorem 1.11
  • Remark 1.12
  • Remark 1.13
  • ...and 61 more