TITLE: What makes Langlands for n=2 easier than Langlands for n>2?
QUESTION [13 upvotes]: I must confess a priori that I haven't read the proof of Taniyama-Shimura, and that my familiarity with Langlands is at best tangential.
As I understand it Langlands for $n=1$ is class field theory. Not an easy theory, but one that was known for a long time.
Langlands for $n=2$ is the Taniyama-Shimura conjecture, proven recently by Andrew Wiles and others (some of whom participate in this forum).
Clearly Taniyama-Shimura required new ideas. What special property of the $n=2$ case made the proof of Taniyama-Shimura possible, that doesn't exist for Langlands with $n\geq 3$?

REPLY [3 votes]: One could (and sometimes would) argue the opposite of your claim/question.  Namely, that "Langlands for $n = 2$" is more difficult than "Langlands for $n > 2$".  Or that more specifically (since I really can't agree with my previous sentence in full generality), "Langlands for $n = 2$" is more difficult than "Langlands for $n > 2$ an odd prime".  Here are two key reasons : 
1) Suppose $F$ is a $p$-adic field, $n$ is prime, and $p$ doesn't divide $n$.  Then the part of the local Langlands correspondence of $GL(n,F)$ dealing with supercuspidal representations (proven by Bushnell/Henniart) is given by $$Ind_{W_E}^{W_F}(\chi) \mapsto \pi(\chi \Delta_{\chi})$$
where $W_F$ is the Weil group of $F$, $E/F$ is a degree $n$ separable extension, $\chi : W_E \rightarrow \mathbb{C}^*$ is a certain type of character ("admissible" to be precise, see Bushnell/Henniart papers/book), $\Delta_{\chi} : W_E \rightarrow \mathbb{C}^*$ is a "twisting character" associated to $\chi$ (again see Bushnell/Henniart), and $\pi(\chi \Delta_{\chi})$ is the supercuspidal representation of $GL(n,F)$ attached to $\chi \Delta_{\chi}$ via the "Howe construction".  If $n$ is an odd prime, then $\Delta_{\chi}$ is either trivial or the unramified quadratic character of $W_E^{ab} \cong E^*$ (note that any character of $W_E$ factors through the abelianization $W_E^{ab}$).  If $n = 2$, then $\Delta_{\chi}$ is much more complicated (see page 217 of Bushnell/Henniart's recent book on $GL(2)$).
2) Let $\pi$ be a supercuspidal representation of $GL(n,F)$ (same assumptions as in the beginning of 1) above).  If $n$ is an odd prime, then the distribution character $\theta_{\pi}$ of $\pi$ is much simpler to write down on elliptic tori than if $n = 2$.  In particular, if $n = 2$, a non-trivial Gauss sum arises in $\theta_{\pi}$, whereas no such term arises if $n$ is an odd prime.  So the theory for $GL(2,F)$ contains some nontrivial arithmetic information that just doesn't arise for $GL(n,F)$, $n$ an odd prime.