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On uniform lower bound of the Galois images associated to elliptic curves

Keisuke Arai
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Abstract

Let p be a prime and K be a number field. Let rho_{E,p}:G_K \longrightarrow Aut(T_p E)\cong GL_2(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, we give an explicit uniform lower bound of the size of the image of rho_{E,p}.

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Keisuke Arai. 2007-03-23. On uniform lower bound of the Galois images associated to elliptic curves. https://arxiv.org/abs/math/0703686

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