# Why doesn't functoriality immediately imply the modularity theorem?

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.

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 $\mathbb{Q}$.

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.

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 $GL(2)$.

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 ex post that there is 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.

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).

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.

[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 $p$.

• The answer to the question in the title is already in your post: you would also need the strong Artin conjecture. More relevant, choosing the varying representations of the finite groups, of necessarily varying dimensions, in an interesting manner, in my eyes, would be a major problem. I can't even think of a way to choose an interesting representation $\rho :GL(E[3])\rightarrow GL(V)$. Sure, I can choose an isomorphism class of representations easily (e.g. $St_1$), but nothing more functorial, which we'd probably want the constructions to be. – Dror Speiser Oct 26 '12 at 18:44
• @ Dror - (a) some cases of functoriality imply some cases of the strong Artin conjecture. Are there cases of the strong Artin conjecture that are not implied by functoriality? – Jonah Sinick Oct 26 '12 at 20:14
• @ Dror - (b) What looks to be arbitrary can sometimes be made canonical by altering one's perspective. For example, there's no canonical square root defined on whole complex plane, but one regains canonicity by enlarging the domain of the square root function to the union of two half split planes glued together. Anyway, as I said in my last paragraph, maybe it's better to look at the (mod $p^k$) presentations as $k$ varies. – Jonah Sinick Oct 26 '12 at 20:25
• Ah sorry, I always forget that 'Langlands Functoriality' means both galois and automorphic side, and not just Automorphic side. – Dror Speiser Oct 26 '12 at 20:59
• So now I'm confused. What do you mean by functoriality? Isn't modularity implied by langlands correspondence using the galois action on the Tate module? – Dror Speiser Oct 26 '12 at 21:22