Codes related to various objects in my thesis

The main object in our work is the image of Galois : if $E/\mathbb{Q}$ is an elliptic curve and $m$ is an integer, and a basis $P,Q$ of $E[m]$, we have the following embedding $$\begin{array}{llll} \rho_{E,m}: & \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) & \rightarrow & \mathrm{GL}_2(\mathbb{Z}/m\mathbb{Z}) \\ & \sigma & \mapsto & \begin{pmatrix} a & c \\ b & d \end{pmatrix}, \end{array} $$ where $a,b,c,d$ are such that $P^\sigma=aP+bQ$ and $Q^\sigma=cP+dQ$. By Serre's open image theorem, $\rho_{E,m}$ is surjective for all integers not divisible by a finite set of primes $\mathcal{L}_E$. Moreover, for all $\ell\in \mathcal{L}_E$, there exists an integer $k\geq 1$ such that $\textrm{Im}(\rho_{E,\ell^k})$ has the same index as $\rho_{E,\ell^k}$ for all $k'\geq k$.

Mazur's program B consists in classifying the elliptic curves $E$ which correspond to a given congruence subgrop of $\text{GL}_2(\mathbb{Z})$. In our paper (available here), we solve the case of subgroups $H$ whose levels are not prime-powers and which occur for infinitely many $j$-invariants.

Data

A complete list of the 1525 ECM-friendly families is available in

We also provide this list in the form of two sage scripts:

Alpha

We pre-computed $\alpha$ (resp. $\alpha_K$ for $K=\mathbb{Q}(\zeta_3)$, $\mathbb{Q}(i)$ and $\mathbb{Q}(\zeta_5)$) for each family n Theorem 4.1 following the method in Sections 5.2 and 5.4 of our article. One can do so for any set corresponding to a subgroup of GL$_2(\mathbb{Z})$ without even knowing if the set is empty, finite or infinite. Given any elliptic curve $E$ with rational coefficients, one tests if its $j$-invariant is coordinate on the modular curve of each subgroup which can occur as Galois image and hence obtains $\alpha(E)$.

Verifications

The following script allows us to verify the Table 26.3.

Several scripts allow to verify the main theore (Theorem 24.7):

Compute torsion point field

  • Ex. 5.14

    Compute Galois image for E and $\ell$

  • Compute Galois image

    Prove surjectivity

  • Alg 10.1 and Alg 10.2

    List of curves

  • Section 11

    Prop. 17.3

  • The polynomial $\mathrm{R}_4$

    Codes for Section 24.2

  • Lifts scripts for Section 24.2.

    Comparison with Morrow

  • Comparison Morrow scripts for Section 26 in Ch. 6.

    Katzian curves

  • Katz scripts scripts for Section 25 in Ch. 6.

    Abelian entanglement

  • Prop 27.3

    I thank my advisor Razvan Barbulescu for his inputs with various codes and generating this html page.