Papers › Open image computations for elliptic curves over number fields
Open image computations for elliptic curves over number fields
David Zywina
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For a non-CM elliptic curve E defined over a number field K, the Galois action on its torsion points gives rise to a Galois representation ρ_E: Gal(K/K)→GL₂(ℤ) that is unique up to isomorphism. A renowned theorem of Serre says that the image of ρ_E is an open, and hence finite index, subgroup of GL₂(ℤ). In an earlier work of the author, an algorithm was given, and implemented, that computed the image of ρ_E up to conjugacy in GL₂(ℤ) in the special case K=ℚ. A fundamental ingredient of this earlier work was the Kronecker-Weber theorem whose conclusion fails for number fields K≠ℚ. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre's theorem for a typical elliptic curve over a fixed number field.
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