Bounds for Serre's open image theorem
Abstract
Let E be an elliptic curve over the rationals without complex multiplication. The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in terms of a Galois representation rhoE:Gal(Qbar/Q) GL2(Zhat). A renowned theorem of Serre says that the image of rhoE is open, and hence has finite index, in GL2(Zhat). We give the first general bounds of this index in terms of basic invariants of E. For example, the index can be bounded by a polynomial function of the logarithmic height of the j-invariant of E. As an application of our bounds, we settle an open question on the average of constants arising from the Lang-Trotter conjecture.
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