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The automorphism group of torsion points of an elliptic curve over a field of characteristic $\ge 5$

Bo-Hae Im, Hansol Kim

TL;DR

This work characterizes the automorphism group of the $p^{e}N$-torsion of the elliptic curve $E_{s,t}: y^{2}=x^{3}+sx+t$ over the function field $K(s,t)$ in characteristic $p\ge 5$, showing it is the full product $(\mathbb{Z}/p^{e}\mathbb{Z})^{\times} \times SL_{2}(\mathbb{Z}/N\mathbb{Z})$ under suitable conditions. The authors develop a two-pronged inductive strategy on $N$: first establishing the base case $N=1$, then proving that if the result holds for $N$, it holds for $N\ell$ for any prime $\ell$ not dividing $pN$ by analyzing the relevant normal subgroup $\mathcal{N}_{p^{e},\mathcal{B}_{N\ell},\ell}$ inside $SL_{2}(\mathbb{Z}/(N\ell)\mathbb{Z})$ and showing it equals the subgroup $\mathfrak{S}_{N\ell}(N)$. The hardest cases are $\ell\in\{2,3\}$ with $\ell\nmid N$, addressed via a discrete-valuation framework that constructs explicit radical extensions with controlled ramification to force the desired subgroup equality. The main result is complemented by a corollary describing the automorphism group over general base fields, using the Weil pairing to connect the determinant to the cyclotomic character. Overall, the paper provides a precise, inductive control of Galois actions on torsion points, with implications for Galois representations attached to elliptic curves in positive characteristic.

Abstract

For a field $\mathbb{K}$ of characteristic $p\ge5$ containing $\mathbb{F}_{p}^{\operatorname{alg}}$ and the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over the function field $\mathbb{K}\left(s,t\right)$ of two variables $s$ and $t$, we prove that for a non-negative positive integer $e$ and a positive integer $N$ which is not divisible by $p$, the automorphism group of the normal extension $\mathbb{K}\left(s,t\right)\left(E_{s,t}\left[p^{e} N\right]\right)$ over $\mathbb{K}\left(s,t\right)$ is isomorphic to $\left(\mathbb{Z}/p^{e}\mathbb{Z}\right)^{\times} \times \operatorname{SL}_{2} \left(\mathbb{Z}/N\mathbb{Z}\right)$.

The automorphism group of torsion points of an elliptic curve over a field of characteristic $\ge 5$

TL;DR

This work characterizes the automorphism group of the -torsion of the elliptic curve over the function field in characteristic , showing it is the full product under suitable conditions. The authors develop a two-pronged inductive strategy on : first establishing the base case , then proving that if the result holds for , it holds for for any prime not dividing by analyzing the relevant normal subgroup inside and showing it equals the subgroup . The hardest cases are with , addressed via a discrete-valuation framework that constructs explicit radical extensions with controlled ramification to force the desired subgroup equality. The main result is complemented by a corollary describing the automorphism group over general base fields, using the Weil pairing to connect the determinant to the cyclotomic character. Overall, the paper provides a precise, inductive control of Galois actions on torsion points, with implications for Galois representations attached to elliptic curves in positive characteristic.

Abstract

For a field of characteristic containing and the elliptic curve defined over the function field of two variables and , we prove that for a non-negative positive integer and a positive integer which is not divisible by , the automorphism group of the normal extension over is isomorphic to .

Paper Structure

This paper contains 9 sections, 23 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Key inductive argument toward the proof of Theorem \ref{['thm:main']}
  3. Proof of $\mathcal{N}_{p^{e},\mathcal{B}_{N \ell},\ell} = \mathfrak{S}_{{N \ell}}\left({N}\right)$
  4. The cases when $\ell \ge 5$ or $\ell \mid N$
  5. The cases when $\ell \in \left\{2,3\right\}$ does not divide $N$
  6. The main ingredient upon a discrete valuation for the proof
  7. Basic properties of valuations
  8. Proof of $\mathcal{N}_{p^{e},\mathcal{B}_{N \ell},\ell} = \mathfrak{S}_{{N\ell}}\left({N}\right)$ for the cases when $\ell \in \left\{2,3\right\}$ does not divide $N$
  9. Proofs of main results

Key Result

Theorem 1.1

For a positive integer $e$, the group homomorphism is an isomorphism.

Figures (1)

  • Figure 1: The diagrams of subfields of $L_{N \ell}$ over $L_{1}$ and their corresponding Galois groups of $L_{N \ell}$ over a field containing $L_{1}$

Theorems & Definitions (45)

  • Theorem 1.1: IK24
  • Theorem 1.2: CF, IK24
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 35 more