The existence of an elliptic curve with a specific Galois representation induced by a character - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:34:27Z http://mathoverflow.net/feeds/question/110882 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110882/the-existence-of-an-elliptic-curve-with-a-specific-galois-representation-induced The existence of an elliptic curve with a specific Galois representation induced by a character Jonah Sinick 2012-10-28T06:40:22Z 2012-10-28T10:38:29Z <p>In Kevin Buzzard's <a href="http://www2.imperial.ac.uk/~buzzard/maths/research/papers/survey.pdf" rel="nofollow">survey article on potential modularity</a> Buzzard writes:</p> <blockquote> <p>Let us say that we have an elliptic curve $E$ over a totally real ﬁeld $F$, and we want to prove that $E$ is potentially modular (that is, that $E$ becomes modular over a ﬁnite extension ﬁeld $F^{′}$ of $F$, also assumed totally real). Here is a strategy. Say $p$ is a large prime such that $E[p]$ is irreducible. Let us write down a random odd $2$-dimensional mod $ℓ$ Galois representation $\rho_{ℓ} : Gal(\overline{F}/F) → GL(2,\mathbf{F}_ℓ )$ which is induced from a character; because this representation is induced it is known to be modular. Now let us consider the moduli space parametrising elliptic curves $A$ equipped with </p> <ol> <li>An isomorphism $A[p] \cong E[p]$</li> <li>An isomorphism $A[ℓ]\cong ρ_ℓ$</li> </ol> <p>This moduli problem will be represented by some modular curve, whose connected components will be twists of $X(pℓ)$ and hence, if $p$ and $ℓ$ are large, will typically have large genus. However, such a curve may well still have lots of rational points, as long as I am allowed to look for such things over an arbitrary ﬁnite extension $F^{′}$ of $F$ !</p> </blockquote> <p>It's not immediately obvious to me that there's an elliptic curve $A$ over some $F^{′}$ satisfying the second condition alone (never mind satisfying both conditions simultaneously). Is there a simple explanation for why there should be such an $A$? Did Professor Buzzard mean "consider the set of A such that $A[ℓ]\cong ρ_ℓ$ for <em>some</em> representation induced by a character" (as opposed to a particular one)?</p> http://mathoverflow.net/questions/110882/the-existence-of-an-elliptic-curve-with-a-specific-galois-representation-induced/110891#110891 Answer by David Loeffler for The existence of an elliptic curve with a specific Galois representation induced by a character David Loeffler 2012-10-28T09:46:46Z 2012-10-28T10:38:29Z <p>In this context, if $\rho$ is a mod $\ell$ representation of $Gal(\overline{F} / F)$, and $A$ is an elliptic curve over an extension $F' / F$, then the statement "$A[\ell] \cong \rho$" needs a little bit of interpretation, because the two sides are representations of different things: $A[\ell]$ is a mod $\ell$ representation of the subgroup $Gal(\overline{F} / F') \subset Gal(\overline{F} / F)$. So the statement is to be read as "$A[\ell]$ is isomorphic as a $Gal(\overline{F} / F')$-representation to the <em>restriction</em> of $\rho$". Now, the bigger $F'$ is, the weaker this condition becomes: in particular, if we take <em>any</em> elliptic curve $A$ over $F$ and define $F'$ to be the extension of $F$ generated by the $\ell$-torsion points of $A$ and the splitting field of $\rho$, then the statement is automatic (both sides are the trivial representation). </p> <p>(This is kind of a stupid example, but maybe you can believe now that there exist non-stupid examples as well!) </p>