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Models of CM elliptic curves with a prescribed $\ell$-adic Galois image

Enrique González-Jiménez, Álvaro Lozano-Robledo, Benjamin York

TL;DR

The paper solves the inverse problem for CM elliptic curves with CM by orders of class number $1$ or $2$: given a prime $\ell$, it determines all $\ell$-adic Galois images that can occur and identifies the twists of a chosen CM model realizing each image, using an explicit labeling system and detailed division-field analysis. The approach builds on established mod-$\ell$ classifications (Zywina) and $\ell$-adic image results (Lozano-Robledo), together with explicit Weierstrass models and a constructive twist-tracking method, aided by Magma computations. Special attention is given to the division fields of CM curves at small CM j-values ($j=0,1728$) and to the $2$- and $3$-adic phenomena, with comprehensive tables listing all possibilities and the corresponding twists. The results yield concrete, verifiable data that facilitate both theoretical exploration and computational verification of CM Galois images, and they provide explicit models and twists suitable for further arithmetic investigations and algorithmic applications.

Abstract

For each prime number $\ell$ and for each imaginary quadratic order of class number one or two, we determine all the possible $\ell$-adic Galois representations that occur for any elliptic curve with complex multiplication by such an order over its minimal field of definition, and then we determine all the isomorphism classes of elliptic curves that have a prescribed $\ell$-adic Galois representation.

Models of CM elliptic curves with a prescribed $\ell$-adic Galois image

TL;DR

The paper solves the inverse problem for CM elliptic curves with CM by orders of class number or : given a prime , it determines all -adic Galois images that can occur and identifies the twists of a chosen CM model realizing each image, using an explicit labeling system and detailed division-field analysis. The approach builds on established mod- classifications (Zywina) and -adic image results (Lozano-Robledo), together with explicit Weierstrass models and a constructive twist-tracking method, aided by Magma computations. Special attention is given to the division fields of CM curves at small CM j-values () and to the - and -adic phenomena, with comprehensive tables listing all possibilities and the corresponding twists. The results yield concrete, verifiable data that facilitate both theoretical exploration and computational verification of CM Galois images, and they provide explicit models and twists suitable for further arithmetic investigations and algorithmic applications.

Abstract

For each prime number and for each imaginary quadratic order of class number one or two, we determine all the possible -adic Galois representations that occur for any elliptic curve with complex multiplication by such an order over its minimal field of definition, and then we determine all the isomorphism classes of elliptic curves that have a prescribed -adic Galois representation.
Paper Structure (25 sections, 27 theorems, 52 equations, 5 figures)

This paper contains 25 sections, 27 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Structure of the article
  3. Code
  4. Preliminaries
  5. Labels
  6. Weierstrass models of elliptic curves with CM by orders of class number 1 and 2
  7. Classification of mod-$\ell$ Galois representation of elliptic curve with CM over $\mathbb Q$
  8. Classification of $\ell$-adic images
  9. The $4$- and $8$-division fields
  10. Orders of class number $1$
  11. The case of $\Delta_K f^2\neq -3$ and $\ell\geq 3$
  12. $\ell\geq 3$ does not divide $\Delta_K f$
  13. $\ell\geq 3$ divides $\Delta_K f$
  14. The case of $\Delta_K f^2\neq -3$ and $\ell=2$
  15. $\Delta_K f^2 \neq -3,-4,-8,-16$ and $\ell =2$
  16. ...and 10 more sections

Key Result

Proposition 2.1

Let $E$ be a CM elliptic curve defined over $\mathbb Q$ with $j_E\neq 0$. The ring of endomorphisms of $E/{\overline{\mathbb{Q}}}$ is an order $\mathcal{O}_{K,f}$ of conductor $f$ in the ring of integers of an imaginary quadratic field of discriminant $\Delta_K$. Take any odd prime $\ell$.

Figures (5)

  • Figure 1: The labels for the $2$-adic images of curves $y^2=x^3+dx$. For each value of $d$ indicated in the second row of each box, the $2$-adic label starts with 2.2.ns7- and ends with the digits indicated in the first row of the box.
  • Figure 2: The labels for the $2$-adic images of curves $y^2=x^3 - 4320d^2 x + 96768d^3$ with $j=8000$. For each value of $d$ indicated in the second row of each box, the $2$-adic label starts with 2.3.ns7- and ends with the digits indicated in the first row of the box.
  • Figure 3: The labels for the $2$-adic images of curves $y^2=x^3-11d^2x+14d^3$ with $j=287496$. For each value of $d$ indicated in the second row of each box, the $2$-adic label starts with 2.4.ns7- and ends with the digits indicated in the first row of the box.
  • Figure 4: The labels for the $2$-adic images of curves $y^2=x^3+16d$ with $j=0$. For each value of $d$ indicated in the second row of each box, the $2$-adic label starts with 2.0.ns5- and ends with the digits indicated in the first row of the box.
  • Figure 5: The labels for the $3$-adic images of curves $y^2=x^3+16d$ with $j=0$. For each value of $d$ indicated in the second row of each box, the $3$-adic label starts with 3.1.ns5- and ends with the digits indicated in the first row of the box.

Theorems & Definitions (53)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Proposition 2.1: zywina, Prop. 1.14
  • Proposition 2.2: zywina, Prop. 1.15
  • Proposition 2.3: zywina, Prop. 1.16
  • Theorem 2.4: lozano-galoiscm, Theorems 1.1 and 1.2
  • Theorem 2.5: lozano-galoiscm, Theorem 1.2
  • Theorem 2.6: lozano-galoiscm, Theorems 1.4 and 1.8
  • proof
  • ...and 43 more