Papers › Possible indices for the Galois image of elliptic curves over Q
Possible indices for the Galois image of elliptic curves over Q
David Zywina
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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 ρ_E : G →GL₂(ℤ̂), where G is the absolute Galois group of the rationals. A well-known theorem of Serre says that the image of ρ_E is open and hence has finite index in GL₂(ℤ̂). 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/ℚ is a non-CM elliptic curve with j-invariant not in $ J$ and with surjective mod ℓ representations for all ℓ>37 (which conjecturally always holds), then the index [GL₂(ℤ̂) : ρ_E(G)] lies in the set I:= {[ 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 ]}. Moreover, I is the minimal set with this property.
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