Why doesn't functoriality immediately imply the modularity theorem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:26:30Z http://mathoverflow.net/feeds/question/110737 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110737/why-doesnt-functoriality-immediately-imply-the-modularity-theorem Why doesn't functoriality immediately imply the modularity theorem? Jonah Sinick 2012-10-26T08:24:18Z 2012-10-27T21:25:39Z <p>Let $E/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, the prime indexed coefficients of its $L$-function agree with those of a weight $2$ cusp eigenform $f$ with integer coefficients. This immediately imply that the coefficients are congruent (mod $p^k$) for every $k > 0$. However, the converse is also true: if the coefficients of the $L$-series of $E$ and that of $f$ are congruent (mod $p^k$) for every prime $k > 0$, then the $L$-series agree.</p> <p>The work of Langlands and Tunnell can be used to show that if the elliptic curve has irreducible (mod $3$) Galois representation, then coefficients of the $L$-function agree with those of a weight $2$ cusp form with coefficients in some algebraic number field $K$ (mod $v$), where $v$ is a prime above $3$ in $K$. This was the starting point of Wiles' proof of the modularity theorem for semi-stable elliptic curves over $\mathbb{Q}$.</p> <p>One could try to get congruence of coefficients (mod $p^k$) for some value of $p$ and for larger values of $k$ by a method analogous to the method via Langlands and Tunnell rather than as a consequence of modularity lifting theorems. One immediately runs into a stumbling block because when $k$ or $p$ is big enough the group is nonsolvable and the methods of Langlands and Tunnell can't be applied (in a known way) to prove relevant cases of the strong Artin conjecture.</p> <p>Nevertheless, there exists an $n$ such that there is an injective representation $\rho: GL(2, \mathbb{Z}/p^k\mathbb{Z}) \to GL(n, \mathbb{C})$. If one can take this representation to be irreducible, then according to the strong Artin conjecture, its $L$-function should be automorphic. Even assuming that this is the case, it's not at all immediately clear (at least, without knowing the modularity theorem) that the $L$-function of the corresponding automorphic representation is related to that a weight $2$ holomorphic cusp eigenform for $GL(2)$. </p> <p>But functoriality can sometimes be used to relate $L$-functions for automorphic representations on one group to $L$-functions of automorphic representations on another group. The arrows only go one way, and in this case it looks like the wrong way, but sometimes one can characterize the arrows' images. Given that we know <em>ex post</em> that there <em>is</em> a relationship between the (mod $p^k$) Galois representation attached to an elliptic curve and that of $f$ (uniform over $k$!), one can ask whether one can see the relationship directly'' from functoriality, without passing through the modularity lifting theorems.</p> <p>Moreover, if one could do this for infinitely many $k$, perhaps one could show that the coefficients of the $L$-function of the elliptic curve $E$ match up with those of the associated modular form $f$ in characteristic $0$ so as to obtain a different proof of the modularity theorem (conditional, of course, on functoriality).</p> <p>The picture that I've sketched above is full of holes like Swiss cheese and it will take me years to understand the precise statements that I allude to above (never mind the proofs!). Nevertheless, I feel that there's a kernel of a well-defined question in what I write. I assume that the strategy that I allude to breaks down somewhere, because otherwise I would have heard about it. I would be grateful to anybody who would be willing to enlighten me as to what goes wrong.</p> <p>[Edited 10/27/12 to incorporate my last paragraph.] One could also attempt to get a result as $p$ varies rather than $k$ if there are technical problems that come up with varying $k$ but not with varying $p$.</p>