# What are the pillars of Langlands?

Since then, I've read quite a bit more (but not nearly enough) and I have a few follow up questions about the big picture. As you will soon see, I'm confused about how to think about things, and seeing the big picture will help me a lot in learning the specifics (learning in the dark is difficult!).

As I understand it, the story goes like this. First, one defined for every number field $\zeta_K(s)=\sum_{\mathfrak{a}} \frac{1}{(N\mathfrak{a})^s}$. One then defines a Dirichlet character, and for any such one defines $L(\chi,s)$. Further, for any $1$-dimensional Galois representation, $\rho: Gal(K/\mathbb{Q}) \rightarrow \mathbb{C}$, one defines $L(\rho,s)$. Now, in the $1$-dimensional case, the main two theorems that comprise class field theory are: if $K$ is abelian over $\mathbb{Q}$ with group $G$, then $\zeta_K(s)=\prod_{\rho \in \hat{G}} L(\rho,s)$; and for any such $\rho$ there exists a unique primitive Dirichlet character $\chi$ such that $L(\rho,s)=L(\chi,s)$. So far I follow the story perfectly.

There is also the issue of what if the base field is not $\mathbb{Q}$, which, admittedly, I don't fully have down.

Already in dimension $2$ I have a hard time figuring out what generalizes what corresponding thing from dimension $1$. For Galois representations, one continues to define $L(\rho,s)$ in a similar manner: as the product over $p$ of the characteristic polynomials of the action of the corresponding Frobenius (whenever defined for that $p$! It is still a little murky to me what happens at the bad primes). But now we have modular forms coming in to the picture, and the whole theory of modular curves. So how does this fit in as a generalization of the 1-dimensional case? Here's my best guess, you can tell me if I'm right. For a modular form $f$, one defines the $L$-function for it by the $q$-expansion of $f$: If $f(z)=\sum a(n)e(nz)$ then $L(f,s)=\sum \frac{a(n)}{n^s}$. Then various things that I do not fully understand come into play, claiming things like: $L(s,f)=\prod_{q|N}(1-a(q)q^{-s})^{-1} \prod_{p\not |N} (1-a(p)p^{-s}+f(p)p^{k-1-2s})^{-1}$ (probably just for $f$'s with some property, akin to being primitive). It seems (is this true?) that Hecke theory implies that these $L$'s are nice'' in the sense that they generalize Dirichlet $L$ functions. Is this the right way to see it? How? What is the $1$-dimensional analogue of modular functions, and modular curves?

Then I imagine that one has the modularity theorem, one of whose versions is(?) that for every $2$-dimensional Galois representation there's a modular function for which $L(\rho,s)=L(f,s)$.

You will notice that at no point did I talk about the adelic aspect. This is because I don't know where to put it. Is the adelic side easily equivalent to the (Dirichlet characters)-(modular functions) side?(are these two even on the same side?) Is it another pillar with which equivalence is far from trivial with both the Galois representations side AND the (Dirichlet characters)-(modular functions) side? In short -- I'm not sure what the pillars of Langlands are!

Further, let us assume that we have some version of Langlands. Is there a conjectural equivalent form of $\zeta_K(s)=\prod_{\rho \in \hat{G}} L(\rho,s)$ for $K/\mathbb{Q}$ not abelian?