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In Abelian l-adic Representations and Elliptic Curves (1968), J. P. Serre showed that the adelic representation $\rho_{E}\colon$\rho_{E}\colon G_K \to \mathrm{GL}2(\hat{\mathbb{Z}})$mathrm{GL}(\hat{\mathbb{Z}}^2)$$associated to an elliptic curve E/K over a number field K has open image. To do it, he uses Shafarevich's Theorem on the finiteness of isomorphism classes of elliptic curves in a given isogeny class to show that the \ell-adic representation \rho{E,\ell}\colon \rho_{E,\ell}\colon G_K \to \mathrm{GL}2(\mathbb{Z}\ell) mathrm{GL}(T_\ell(E))$$ is irreducible for all$\ell$and that the mod$\ell$representation$\bar{\rho}_{E,\ell}\colon $\bar{\rho}_{E,\ell}\colon G_K \to \mathrm{GL}2(\mathbb{F}\ell)$ mathrm{GL}(E[\ell]) is irreducible for almost all $\ell$.

My question is, do we now have a method of proving this theorem without using Shafarevich's Theorem? The latter depends on Siegel's Theorem, which depends on Roth's Theorem in Diophantine Geometry.

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# Serre's Open Image Theorem Without Shafarevich's Theorem

In Abelian l-adic Representations and Elliptic Curves (1968), J. P. Serre showed that the adelic representation $\rho_{E}\colon G_K \to \mathrm{GL}2(\hat{\mathbb{Z}})$ associated to an elliptic curve $E/K$ over a number field $K$ has open image. To do it, he uses Shafarevich's Theorem on the finiteness of isomorphism classes of elliptic curves in a given isogeny class to show that the $\ell$-adic representation $\rho{E,\ell}\colon G_K \to \mathrm{GL}2(\mathbb{Z}\ell)$ is irreducible for all $\ell$ and that the mod $\ell$ representation $\bar{\rho}_{E,\ell}\colon G_K \to \mathrm{GL}2(\mathbb{F}\ell)$ is irreducible for almost all $\ell$.

My question is, do we now have a method of proving this theorem without using Shafarevich's Theorem? The latter depends on Siegel's Theorem, which depends on Roth's Theorem in Diophantine Geometry.