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On uniform lower bound of the Galois images associated to elliptic curves. (English) Zbl 1211.11066

Summary: Let \(p\) be a prime and let \(K\) be a number field. Let \(\rho_{E,p}: G_K \to\operatorname{Aut}(T_pE)\cong \text{GL}_2(\mathbb Z_p)\) be the Galois representation given by the Galois action on the \(p\)-adic Tate module of an elliptic curve \(E\) over \(K\). Serre showed that the image of \(\rho_{E,p}\) is open if \(E\) has no complex multiplication. For an elliptic curve \(E\) over \(K\) whose \(j\)-invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of \(\rho_{E,p}\).

MSC:

11G05 Elliptic curves over global fields
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References:

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