Possible indices for the Galois image of elliptic curves over Q

Abstract

For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation ${\rho }_E:G\to G{L}_2(\hat{\mathbb{Z}})$, where $G$ is the absolute Galois group of the rationals. A well-known theorem of Serre says that the image of ${\rho }_E$ is open and hence has finite index in $G{L}_2(\hat{\mathbb{Z}})$. We will study what indices are possible assuming that we are willing to exclude a finite number of possible $j$-invariants from consideration. For example, we will show that there is a finite set $J$ of rational numbers such that if $E/\mathbb{Q}$ is a non-CM elliptic curve with $j$-invariant not in $J$ and with surjective mod $\ell$ representations for all $\ell >37$ (which conjecturally always holds), then the index $[G{L}_2(\hat{\mathbb{Z}}):{\rho }_E(G)]$ lies in the set \[ I:= \left\{\begin{array}{c}2, 4, 6, 8, 10, 12, 16, 20, 24, 30, 32, 36, 40, 48, 54, 60, 72, 84, 96, 108, 112,120, 144, \\192, 220, 240, 288, 336, 360, 384, 504, 576, 768, 864, 1152, 1200, 1296, 1536 \end{array}\right\}. \] Moreover, $I$ is the minimal set with this property.

Minor corrections and updates. New permanent link to related code with improvements in code (it was running extraordinarily slow on current versions of Magma)

Cited by in corpus (2)

• Torsion subgroups of rational elliptic curves over the compositum of all cubic fields
• Congruences of elliptic curves arising from non-surjective mod $N$ Galois representations