# 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 $$\rho_{E}\colon G_K \to \mathrm{GL}(\hat{\mathbb{Z}}^2)$$ 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.

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This is an interesting question and I suspect there is a way using p-adic Hodge theory. In the meanwhile, I thought I'd point out that Shafarevich's theorem here requires only Siegels theorem for the `discriminant elliptic curves,' something like 4a3−27b2=c. These have CM, and a proof of finiteness that doesn't use Diophantine approximations at all can be found in annals.math.princeton.edu/wp-content/.../annals-v172-n1-p16-p.pdf (By the way, I had mistakenly posted this as an answer earlier.) –  Minhyong Kim Jul 18 '12 at 22:28
Oh sorry, I should also say that the remark above applies only to elliptic curves over $\mathbb{Q}$. –  Minhyong Kim Jul 18 '12 at 22:29
@Davidac897: I asked a somewhat similar question a while back out my own ignorance of Falting's work: mathoverflow.net/questions/37212 . I wanted to deduce Shafarevich's theorem over $\Bbb Q$ from modularity without using Siegel's Theorem, but my argument was cyclic because I unknowingly assumed Tate's Isogeny Conjecture, which was proved by Faltings by proving Shafarevich in all dimensions. It was mentioned in the comments there that you can deduce Siegel's Theorem from Faltings Theorem (Mordell's Conjecture) which doesn't use Diophantine Approximation. Not sure if that will help you. –  Jamie Weigandt Jul 19 '12 at 0:09

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.
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.
I originally said without Shafarevich, though what I implicitly hoped for was something without using Sigel's Theorem either. It sounds a bit overkill to go to Faltings's Theorem, but I wonder whether Faltings's proof is simpler in dimension $1$. –  David Corwin Jul 19 '12 at 22:14