and line-transitive linear spaces.

Gill, Nick

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Let $p$ be a prime and let $K$ be a number field. Let $ρ_{E, p} : G_{K} \to Aut (T_{p} E) ≅ {GL}_{2} (ℤ_{p})$ 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 $ρ_{E, p}$ 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 $ρ_{E, p}$ .

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