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The automorphism group of the $p^{n}$-torsion points of an elliptic curve over a field of characteristic $p \ge 5$

Bo-Hae Im, Hansol Kim

Abstract

For a field $K$ of characteristic $p\ge5$ and the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over the function field $K\left(s,t\right)$ of two variables $s$ and $t$, we prove that for a positive integer $n$, the automorphism group of the normal extension $K\left(s,t\right)\left(E_{s,t}\left[p^{n}\right]\right)/K\left(s,t\right)$ is isomorphic to $\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{\times}$, and its inseparable degree is $p^{n}$.

The automorphism group of the $p^{n}$-torsion points of an elliptic curve over a field of characteristic $p \ge 5$

Abstract

For a field of characteristic and the elliptic curve defined over the function field of two variables and , we prove that for a positive integer , the automorphism group of the normal extension is isomorphic to , and its inseparable degree is .

Paper Structure

This paper contains 5 sections, 17 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Supersingular $j$-invariants and division polynomials
  3. Supersingular $j$-invariants
  4. Division polynomials and the formula for the $x$-coordinates under the multiplication-by-$m$ map
  5. Proofs of Theorem \ref{['thm:main']}, Corollary \ref{['cor:main1']}, and Corollary \ref{['cor:main2']}

Key Result

Theorem 1.1

Let $K$ be a field of characteristic $p \ge 5$. For the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over $K\left(s,t\right)$, the automorphism group of $K\left(s,t\right)\left(E_{s,t}\left[p^{n}\right]\right)$ fixing $K\left(s,t\right)$ is isomorphic to $\left({\mathbb{Z}}/p^{n}{\mathbb

Figures (1)

  • Figure 1: The tower of subfields of $K\left(s,t\right)\left(E_{s,t}\left[p^{n}\right]\right)$ containing $K\left(s,t\right)$

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1: Silverman
  • Remark 2.2
  • Lemma 2.3: Silverman
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 24 more