Serre's Open Image Theorem Without Shafarevich's Theorem

2012-07-18T21:51:30Z

In Abelian l-adic Representations and Elliptic Curves (1968), J. P. Serre showed that the adelic representation $$\rho_{E}\colon G_K \to \mathrm{GL}(\hat{\mathbb{Z}}^2)$$ 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 $$\rho_{E,\ell}\colon G_K \to \mathrm{GL}(T_\ell(E))$$ is irreducible for all $\ell$ and that the mod $\ell$ representation $$\bar{\rho}_{E,\ell}\colon G_K \to \mathrm{GL}(E[\ell])$$ 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.

Answer by DZ for Serre's Open Image Theorem Without Shafarevich's Theorem

2012-07-19T17:30:16Z

Masser and Wüstholz have given an effective proof that the representation $\bar{\rho}_{E,\ell}\colon G_K \to \mathrm{GL}(E[\ell])$ is irreducible for all $\ell$ greater than some constant $c_E$, see their paper Some effective estimates for elliptic curves. They use isogeny bounds coming from transcendence theory to prove Shafarevich's Theorem without Siegel's theorem. They show that $c_E$ can be chosen to be less than $C h^4$ where $h$ is some naive height attached to $E/K$ and $C$ is a constant that can in principle be computed.

(The isogeny bounds have since been repeated improved. The state of the art might be the paper Théorème des périodes et degrés minimaux d'isogénies of Gaudron and Rémond.)

Added afterwards: The surjectivity of $\bar{\rho}_{E,\ell}$ for $\ell$ sufficiently large is also discussed by Masser and Wüstholz in Galois properties of division fields of elliptic curves. It is effective and again does not require Siegel's theorem.

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Serre's Open Image Theorem Without Shafarevich's Theorem

Answer by DZ for Serre's Open Image Theorem Without Shafarevich's Theorem