Images of the 2-adic Galois representation attached to elliptic curves

Jeremy Rouse and David Zureick-Brown

In our paper "Elliptic curves over $\mathbb{Q}$ and 2-adic images of Galois" (on arXiv here), we classify the possible images of the 2-adic Galois representations attached to non-CM elliptic curves $E/\mathbb{Q}$. This website serves as a companion to the paper, and includes links to websites documenting the relevant lattice of modular curves, as well as files with data, computation logs, and Magma scripts that are necessary for our results.

The lattice of modular curves can be accessed at the link for $X_{1}$.

Facts about the subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$, the corresponding modular curves, and the rational points on them:

• We call a subgroup of $H \subset {\rm GL}_{2}(\mathbb{Z}_{2})$ arithmetically maximal if $\det \colon H \to \mathbb{Z}_{2}^{\times}$ is surjective, there is an $M \in H$ with determinant $-1$ and trace zero, and there is no subgroup $K$ with $H \subseteq K$ so that the modular curve $X_{K}$ has genus $\geq 2$.
• There are $727$ arithmetically maximal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$ that contain $-I$ (up to conjugacy in ${\rm GL}_{2}(\mathbb{Z}_{2})$), and each of them has a webpage (which can be found from the link above).
• The curves $X_{H}$ have genus $0$, $1$, $2$, $3$, $5$ or $7$. We find all the rational points on each $X_{H}$.
• Each curve $X_{H}$ has good reduction away from $2$.
• We compute a model for each $X_{H}$ unless there is a subgroup $K$ with $H \subset K$ so that $X_{K}$ has finitely many rational points. In total, we compute models of $345$ curves $X_{H}$ for subgroups $H$ containing $-I$. Of these, $185$ have genus zero, $58$ have genus one, $57$ have genus two, $22$ have genus three, $20$ have genus five, and $4$ have genus seven.
• If $H$ is a subgroup that contains $-I$ and $X_{H}$ has any non-cuspidal, non-CM rational points, and there is a subgroup $K \subseteq H$ with $-I \not\in K$ then (i) $X_{H} \cong \mathbb{P}^{1}$, and (ii) we compute a model for the universal elliptic curve $E/U$, where $U$ is the locus of $X_{H}$ with $j \ne 0$, $j \ne 1728$, and $j \ne \infty$. (Each of the $1006$ subgroups $K$ of this type also has a webpage.)
• If $E/\mathbb{Q}$ is a non-CM elliptic curve, the image of $\rho_{E} : {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\rm GL}_{2}(\mathbb{Z}_{2})$ has index equal to $1$, $2$, $3$, $4$, $6$, $8$, $12$, $16$, $24$, $32$, $48$, $64$ or $96$.
• For a non-CM curve $E$, the image of $\rho_{E}$ in ${\rm GL}_{2}(\mathbb{Z}_{2})$ always contains the kernel of the map ${\rm GL}_{2}(\mathbb{Z}_{2}) \to {\rm GL}_{2}(\mathbb{Z}/32\mathbb{Z})$.
• Of the $1733$ subgroups $H$ we list, there are $1215$ that have non-cuspidal, non-CM rational points.
• There are $1208$ possibilities for the image of $\rho_{E} \colon {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\rm GL}_{2}(\mathbb{Z}_{2})$.
• The seven curves $X_{150}$, $X_{153}$, $X_{155}$, $X_{156}$, $X_{165}$, $X_{166}$, and $X_{167}$ are all modular curves $X_{H}$ that are elliptic with positive rank, and every rational point on $X_{H}$ lifts to a rational point on $X_{K}$ for some index $2$ subgroup $K$ of $H$.
• Every rational point on a curve $X_{H}$ of genus one that has rank zero is a cusp or a CM point.
• The only genus $2$ curve with non-cuspidal, non-CM rational points is $X_{441}$, also known as $X_{ns}^{+}(16)$. This curve has two non-cuspidal, non-CM rational points.
• The genus $3$ curves $X_{556}$, $X_{558}$, $X_{563}$, and $X_{566}$ are hyperelliptic.
• The only genus $3$ curves with non-cuspidal, non-CM rational points are $X_{556}$, $X_{558}$, $X_{563}$, $X_{566}$, $X_{619}$, and $X_{649}$. Each of these gives rise to a single $j$-invariant.
• The genus $5$ and $7$ curves are represented (on this website) by a singular model that is a double cover of an elliptic curve. Non-singular canonical models are given below in the log file "desing3.out".
• All the rational points on the genus $5$ and $7$ curves are either cusps or CM points.
• There are many instances of non-conjugate subgroups $H$ and $K$ with $X_{H} \cong X_{K}$. Within the $22$ genus three curves, there are at most $7$ isomorphism classes. Within the $20$ genus five curves, there are at most $10$ isomorphism classes. The $4$ genus seven curves, fall into two isomorphism classes.
• The curve $X_{504}$ is a genus two curve with six rational points. Its Jacobian has rank zero, and torsion subgroup $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/10\mathbb{Z}$.
• If $E/\mathbb{Q}$ is a non-CM elliptic curve whose mod $2$ image has index $1$, the $2$-adic image can have index as large as $64$.
• If $E/\mathbb{Q}$ is a non-CM elliptic curve whose mod $2$ image has index $2$, the $2$-adic image has index $2$ or $4$.
• If $E/\mathbb{Q}$ is a non-CM elliptic curve whose mod $2$ image has index $3$, the $2$-adic image can have index as large as $96$.
• If $E/\mathbb{Q}$ is a non-CM elliptic curve whose mod $2$ image has index $6$, the $2$-adic image can have index as large as $96$, although some quadratic twist of $E$ must have $2$-adic image with index less than $96$.

Below is a table with links to data files.

Below is a table with links to log files.

Below is a table with links to program files.