Example of a diophantine application of an open image theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:04:30Z http://mathoverflow.net/feeds/question/109298 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109298/example-of-a-diophantine-application-of-an-open-image-theorem Example of a diophantine application of an open image theorem Adam Harris 2012-10-10T13:58:43Z 2012-10-11T02:02:25Z <p>I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model theoretically nice e.g. stable). </p> <p>So I have some idea as to why these open image theorems should hold from a model theoretic viewpoint, and I know that these are regarded as important theorems, but I don't think I've ever come across a diophantine application of an open image theorem in the literature and I'd like to see one.</p> <p>I'm most familiar with Serre's open image theorem for elliptic curves so an example in this context would be ideal.</p> http://mathoverflow.net/questions/109298/example-of-a-diophantine-application-of-an-open-image-theorem/109301#109301 Answer by Barinder Banwait for Example of a diophantine application of an open image theorem Barinder Banwait 2012-10-10T14:52:51Z 2012-10-10T14:52:51Z <p>Serre's open image theorem (on page IV-20 in his book "Abelian $l$-adic representations...) for non-CM elliptic curves $E/K$ is equivalent to the statement that, for almost all $l$ (how large depending on $E$ and $K$), the $l$-adic representation attached to $T_l(E)$ is surjective. Ditto for the mod-$l$ representation. These are nice examples of number theoretic applications. Explicit bounds are also known (work of Hall, Cojocaru and others...). Note that how large $l$ must be is expected to be independent of $E$, and should depend only on $K$. For example, if $K = \mathbb{Q}$, it is hoped that 37 is large enough for any non-CM elliptic curve. </p> <p>It is conjectured that, for $A/K$ any abelian variety for which $End_{\overline{K}}(A) = \mathbb{Z}$, there should be a similar open-image theorem. This is known (by work of Serre) when the dimension of $A$ is 2,6, or odd. In particular, for such an $A/K$, and for sufficiently large $l$, the mod-$l$ representation has image $GSp_{2g}(\mathbb{F}_l)$. This is also nice.</p> <p>(Work of Bogomolov says that the $l$-adic image of $A$ is open (with the $l$-adic topology) in <code>$G_{A,l}(\mathbb{Q}_l)$</code> ; here $G_{A,l}$ is the $l$-adic algebraic monodromy group. See <a href="http://www.martinorr.name/blog/2010/11/27/images-of-galois-representations" rel="nofollow">this blog post of Martin Orr</a> for a discussion of these groups.)</p> http://mathoverflow.net/questions/109298/example-of-a-diophantine-application-of-an-open-image-theorem/109317#109317 Answer by Kevin Ventullo for Example of a diophantine application of an open image theorem Kevin Ventullo 2012-10-10T18:41:41Z 2012-10-10T18:51:37Z <p>Well, this isn't explicitly diophantine, but here goes:</p> <p>If $f$ is a level one weight $k$ eigenform with rational coefficients, the image of the attached Galois representation </p> <p>$\rho_f:G_{\mathbb{Q}}\rightarrow GL_2(\hat{\mathbb{Z}})$ </p> <p>is open in the subgroup $G$ defined by demanding </p> <p>$det(G)\subset \hat{\mathbb{Z}}^{\times{k-1}}$. </p> <p>In particular, the image contains an open subgroup of $SL_2(\hat{\mathbb{Z}})$. This has the following arithmetic consequence:</p> <p><em>For almost all prime numbers $p$, there exists a non-solvable Galois extension $K/\mathbb{Q}$ ramified only at $p$</em>. </p> <p>In fact, Serre shows that for the unique normalized weight 12 level 1 cuspform, the list of exceptional primes is 2,3,5,7,23,691. This theorem is now known for all p, although the last known case, p=7, was resolved only very recently by Dieulefait.</p> http://mathoverflow.net/questions/109298/example-of-a-diophantine-application-of-an-open-image-theorem/109348#109348 Answer by Joe Silverman for Example of a diophantine application of an open image theorem Joe Silverman 2012-10-11T02:02:25Z 2012-10-11T02:02:25Z <p>Here's an application to independence of Heegner points. (But if you search on MathSciNet for papers that reference Serre's two results, I expect you'll find a very large number of applications.)</p> <p>Let $E/\mathbb{Q}$ be an elliptic curve with no CM, and let $\Phi:X_0(N)\to E$ be a modular parametrization. (Wiles et.al. show that $\Phi$ exists for all such $E$.) The modular curve $X_0(N)$ has special points called <em>Heegner points</em> associated to pairs $(C,\Gamma)$, where $C$ is a CM elliptic curve and $\Gamma\subset C$ is a cyclic subgroup of order $N$. More precisely, we can associate to each imaginary quadratic field $K$ (satisfying some conditions) a Heegner point $x_K\in X_0(\overline{\mathbb{Q}})$ associated to the maximal order in $K$.</p> <p><strong>Theorem</strong> [1] Let $K_1,\ldots,K_r$ be distinct imaginary quadratic fields such that the odd parts of their class numbers are sufficiently large. Then the points $\Phi(x_{K_1}),\ldots,\Phi(x_{K_r})$ are linearly independent in the group $E(\overline{\mathbb{Q}})$.</p> <p>The proof uses Serre's image of Galois theorem in a crucial way. Not simply that the image of Galois is open in each $\hbox{Aut}(T_\ell(E))$, but also that it is surjective for almost all $\ell$.</p> <p>[1] M. Rosen, JH Silverman, On the independence of Heegner points associated to distinct quadratic imaginary fields, <em>Journal of Number Theory</em> <strong>127</strong> (2007), 10-36.</p>