Serre's Open Image Theorem Without Shafarevich's Theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T10:00:43Z http://mathoverflow.net/feeds/question/102588 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102588/serres-open-image-theorem-without-shafarevichs-theorem Serre's Open Image Theorem Without Shafarevich's Theorem David Corwin 2012-07-18T21:51:30Z 2012-07-20T00:09:21Z <p>In <em>Abelian l-adic Representations and Elliptic Curves</em> (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 $\ell$.</p> <p>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.</p> http://mathoverflow.net/questions/102588/serres-open-image-theorem-without-shafarevichs-theorem/102687#102687 Answer by DZ for Serre's Open Image Theorem Without Shafarevich's Theorem DZ 2012-07-19T17:30:16Z 2012-07-19T22:26:09Z <p>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 <em>Some effective estimates for elliptic curves</em>. 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.</p> <p>(The isogeny bounds have since been repeated improved. The state of the art might be the paper <em>Théorème des périodes et degrés minimaux d'isogénies</em> of Gaudron and Rémond.)</p> <p>Added afterwards: The surjectivity of $\bar{\rho}_{E,\ell}$ for $\ell$ sufficiently large is also discussed by Masser and Wüstholz in <em>Galois properties of division fields of elliptic curves</em>. It is effective and again does not require Siegel's theorem.</p>