# Connection of Galois representation and arithmetic geometry

This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that people are interested in studying such Galois representations. There are lots of theorems which explain images of such representations are contained in special/open/closed subgroups or, form special subgroups. What geometric information can we get from such theorems? (or, why are such theorems so good for us?)

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Another example is the Tate conjecture for Abelian varieties $A, A'$ over finitely generated fields $K$: $\mathrm{Hom}(A,A')\otimes_{\mathbf{Z}}\mathbf{Z}_\ell = \mathrm{Hom}_{G_K}(T_\ell A,T_\ell A')$ The surjectivity means roughly that you can construct isogenies between Abelian varieties from $G_K$-equivariant homomorphisms between their Tate modules.
Two of the equivalent formulations of modularity of elliptic curves are: 1) The Galois representation is modular [Galois representation]. 2) There is a finite morphism $X_0(N) \to E$ [geometry].
@DavidLoeffler Actually, Neron-Ogg-Shafarevich is very geometric. One should view an abelian variety as a group scheme over a disk (I'm thinking of $A$ over Spec$(R)$ for a DVR $R$) whose fibers, except possibly at the cetnral point, are smooth. Then the ramification of the Galois representation tells you whether the fiber over the central point of the disk is singular or non-singular. [This is the intuition, I'm being a little rough here on details, but they can be filled in.] Of course, here we want to take the image of the inertia group, but that's the image of Gal($K^s/K^{nr}$). – Joe Silverman Jul 12 '14 at 14:37
@JoeSilverman: of course, I would be the last person to dispute that N-O-S tells us very significant geometric information; but as you yourself point out, you need more information to apply it than just knowing the image of $\rho$ as an abstract subgroup of $GL_2(\mathbf{Q}_\ell)$. I thought the OP wanted to know what geometric information we can get from things like Serre's open image theorem, etc, which can be formulated purely as statements about the group $Im(\rho)$. – David Loeffler Jul 12 '14 at 18:13
@DavidLoeffler Okay, I see what you mean. OTOH, the OP just said "image of Galois" without specifying which Galois group, so if one is willing to look at the image of $GL_2(\mathbb{Q}^{nr})$ as an abstract group, the triviality or not of that image will give the condition to apply N-O-S. But if you think that's stretching the OP's question, I can't disagree. – Joe Silverman Jul 12 '14 at 19:09