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Automorphisms of finite fields from isogeny cycles

Kéva Djambaé

Abstract

We develop an explicit geometric construction of automorphisms of finite fields arising from isogeny cycles. Let $k$ be a finite field, $E/k$ an elliptic curve, and $\ell$ an integer coprime to $\mathrm{char}(k)$. Let $\mathfrak{h}$ be an ideal of $\mathrm{End}(E)$ dividing $\ell$, and consider the corresponding torsion subgroup $E[\mathfrak{h}]\subseteq E[\ell]$. From the action of End(E) on $E[\mathfrak{h}]$, we construct the splitting field $K$ of the $x$-coordinates of points in $E[\mathfrak{h}]$ and the associated Galois group $\mathrm{Gal}(K/k)$. This yields $(\mathrm{End}(E)/\mathfrak{h})^* \to \mathrm{Gal}(K/k)$ a group homomorphism.

Automorphisms of finite fields from isogeny cycles

Abstract

We develop an explicit geometric construction of automorphisms of finite fields arising from isogeny cycles. Let be a finite field, an elliptic curve, and an integer coprime to . Let be an ideal of dividing , and consider the corresponding torsion subgroup . From the action of End(E) on , we construct the splitting field of the -coordinates of points in and the associated Galois group . This yields a group homomorphism.
Paper Structure (27 sections, 12 theorems, 41 equations)

This paper contains 27 sections, 12 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Elliptic curves and torsion points
  4. Coordinate Rings and Functoriality
  5. Normal basis
  6. Constructing automorphisms of finite fields from elliptic curves
  7. Case when $\ell$ is a prime
  8. The endomorphism $f$ induces a homomorphism of fields
  9. The endomorphism $f$ induces a Galois automorphism
  10. Construction in the general case
  11. Extension to composite degrees
  12. Main result
  13. Constructing the Galois group
  14. Algorithmic considerations
  15. Constructing an endomorphism of elliptic curves from isogeny cycles
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $k = \mathbb{F}_{q}$ be a finite field of characteristic $p$ and let $\ell$ be a positive integer coprime to $p$. Let $E/k$ be an elliptic curve, and let $\mathfrak{h}$ be an ideal of $\mathop{\mathrm{End}}\nolimits(E)$ of norm $\ell$. Denote by $H=E[\mathfrak{h}]$ the subscheme of $E[\ell]$ def Denote by $\psi$ an irreducible factor of the kernel polynomial $\psi_H$ of $H$ and the degree of $

Theorems & Definitions (33)

  • Theorem 1.1: Main result
  • Proposition 2.1
  • Proposition 2.2
  • Remark 1
  • Proposition 2.3
  • proof
  • Remark 2
  • Lemma 1
  • proof
  • Definition 3.1
  • ...and 23 more