What are the pillars of Langlands?

Question

I had previously asked: Narratives in Modular Curves

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?

Answer 1

In either the one or two dimensional case, there are two sides: Galois and automorphic.

Let me talk for a moment about the $n$-dimensional situation.

The Galois side involves studying continuous $n$-dimensional representations of the Galois group of a number field $K$. There are subtleties here about over what field these representations are defined, which I will return to. For now, let's imagine that they are representations into $GL_n(\mathbb C)$, hence what are usually called Artin representations.

The automorphic side involves so-called automorphic representations of $GL_n(\mathbb A)$, where $\mathbb A$ is the adele ring of $K$. These are irreduible representations which appear as Jordan--Holder factors in the space of automorphic forms, which is, roughly speaking, the space of functions on the quotient $GL_n(K) \backslash GL_n(\mathbb A)$. (To provide some orientation with regard to this notion: Note that, by Frobenius reciprocity, any irreducible representation of a group $G$ appears in the space of functions on $G$ --- here I am ignoring topological issues --- and so any irreducible rep. of $GL_n(\mathbb A)$ will appear in the space of functions on $GL_n(\mathbb A)$. On the other hand, appearing in the space of functions on the quotient $GL_n(K)\backslash GL_n(\mathbb A)$ turns out to be a serious restriction --- most representations of $GL_n(\mathbb A)$ don't appear there.)

The idea --- roughly --- is that $n$-dimensional Galois representations will match with automorphic representations. How does this happen?

Well, $n$-dimensional representations have traces of Frobenius elements, so to each unramified prime we can attach a number. On the other hand, it turns out that automorphic representations have essentially canonical generators, which can be interpreted as Hecke eigenforms, and so (for primes not dividing the conductor of the representation) we get a Hecke eigenvalue. The matching is given by the rule that traces of Frobenius elements should equal Hecke eigenvalues.

Let's consider the case of dimension $n = 1$:

First, any $1$-dim'l rep'n of the Galois group of $K$ factors through $G_K^{ab}$, while the space of functions on the abelian group $K^{\times}\backslash \mathbb A^{\times}$ (which is the adelic quotient considered above in the case $n = 1$) is (by Fourier theory for locally compact abelian groups, and speaking somewhat loosely) the sum of spaces spanned by characters, so that automorphic representations are just characters of $K^{\times}\backslash \mathbb A^{\times}$.

So our goal is to match characters of $G_K^{ab}$ with characters of $K^{\times}\backslash \mathbb A^{\times}$.

Now global class field theory actually does something stronger: it directly relates $G_K^{ab}$ to $K^{\times}\backslash \mathbb A^{\times}$. In particular, finite order characters of the latter (which are just Dirichlet characters in the case $K = \mathbb Q$) correspond to finite order characters of the former, the $L$-functions match, and so on.

Now consider the case $n = 2$:

The quotient $GL_2(K)\backslash GL_2(\mathbb A)$ is not a group, just a space (and this is the same for any $n \geq 2$). Also, there is no quotient of the Galois group $G_K$ akin to $G_K^{ab}$ with the property that all $2$-dimensional reps. of $G_K$ factor precisely through that quotient.

In short, unlike in the case $n = 1$, we can't hope to find groups that will be related in some direct manner (unlike in the case $n = 1$, where class field theory gives a direct relation between $G_K^{ab}$ and $K^{\times}\backslash \mathbb A^{\times}$); the relation really has to be made at the level of representations.

Also, when you consider the quotient $GL_2(K)\backslash GL_2(\mathbb A)$, it is much harder (in fact, impossible) to separate the archimedean and non-archimedean primes. If you play around and follow up some of the references suggested by others, you will see (in the case $K = \mathbb Q$) that this quotient is related to the upper half-plane and congruence subgroups, and the generating vectors for automorphic representations are precisely primitive Hecke eigenforms (either holomorphic modular forms or else Maass forms). The archimedean prime contributes the upper half-plane, and the finite primes contribute the level of the congruence subgroup and the Hecke operators.

Thus the analogue of a Dirichlet character in the $2$-dimensional case is a Hecke eigenform. (But it is better to regard the latter as corresponding more generally to idele class characters --- also called Hecke characters or Grossencharacters; these are not-necessarily-finite-order generalizations of Dirichlet characters.)

If you really want to study just Galois represenations with finite image (i.e. Artin representations), on the automorphic side (for $K = \mathbb Q$ and $n = 2$) you should restrict to weight one holomorphic modular forms and Maass forms with $\lambda = 1/4$. (The former are proved to correspond precisely to two-dimensional reps. for which complex conjugation has non-scalar image, while the latter are conjectured to correspond to two-dimensional reps. for which complex conjugation has scalar image.) The other Maass forms are not supposed to correspond to any Galois representations (just as non-algebraic idele class characters --- those that are not type $A_0$ in Weil's terminology --- don't correspond to anything on the Galois side in class field theory), while higher weight holomorphic modular forms are supposed to corresond to compatible systems of two-dimensional $\ell$-adic Galois representations with infinite image (just as infinite order algebraic Hecke characters correspond to compatible systems of one-dimensional $\ell$-adic Galois representations --- see Serre's bok on this topic).

As the preceding summary shows, the big picture here is pretty big, and the technical details are quite extensive and involved. There is the added complication that the proofs of what is known in the non-abelian case are very involved, and often use methods that don't play a role in the general story, but seem to be crucial for the arguments to go through. (E.g. Mellin transforms don't play any role in the general theory of attaching $L$-functions to automorphic representations, but in the case $n = 2$, the fact that you can pass from a modular form to its $L$-function via a Mellin transform is often useful and important.)

What I would suggest is that you try to learn a little about Hecke characters beyond the finite order case, perhaps from Weil's original article and also from Serre's book. (Silverman's treatment of complex multiplication elliptic curves may also help.) This will help you become familiar with the crucial idea of compatible families of $\ell$-adic Galois representations, and see how it relates to objects on the automorphic side, in the fundamental $n = 1$ case.

Then, to begin to grasp the $n = 2$ case, you will want to understand how modular forms give compatible systems of two-dimensional $\ell$-adic representations. The general statement here is due to Deligne, and can be learnt (at least at first) as a black-box. The case of weight 2, which includes the case of elliptic curves over $\mathbb Q$, is easier, and it's possible to learn the whole story (other than proof of the modularity theorem itself) in a reasonable amount of time. The case of weight one is special, and is treated in a beautiful paper of Deligne and Serre. The converse here (that every two-dimensional Artin rep'n of appropriate type comes from a weight one form) is at least as difficult as the modularity of elliptic curves, and so learning the statement should suffice at the beginning.

And now you confront the difficulties in learning non-abelian class field theory: already in the two-dimensional case, the theory has some aspects that remain almost entirely conjectural, namely the conjectured relationship between certain Maass forms and certain two-dimensional Galois representations. So at this point, you have to choose whether you just want to get some sense of the big picture and the expected story in general (say by looking over Clozel's Ann Arbor article and the recent preprint of Buzzard and Gee), or to begin actually working in the area. If you choose the latter, then the big picture is useful, but only in a limited way: to make progress, it seems that one has to then start learning many techniques that don't seem to be essential to the story from the big picture point of view, but which are currently the only known methods for making progress (depending on what direction you want to pursue, these include the trace formula, Shimura varieties, the deformation theory of Galois representations, the Taylor--Wiles method, ... ).

Answer 2

This is an answer to the part of your question concerning the case of dimension $1$. I'll omit any details in higher dimension; even if you're only vaguely familiar with elliptic curves you will see the bigger picture.

1. A Pell conic is an affine curve of the form $C_N: Q_0(X,Y) = 1$, where $Q_0$ is the principal binary form with discriminant $N$ (if $N = 4m$ with $m \equiv 3 \bmod 4$, then $Q(X,Y) = X^2 - mY^2$). For each prime $p$ not dividing $N$, the curve has a smooth reduction modulo $p$, and the number of points is $p - a_p$ for $a_p = (N/p)$ (the Kronecker symbol).

2. We call $C_N$ modular if there exists a modulus $m$ such that $a_p$ only depends on the residue class of $p$ modulo $m$. It can be shown that $m = N$ always works, and that $(N/p) = \chi(p)$ for a primitive Dirichlet character $\chi$ with conductor $N$.

3. We attach a "Pell form" $$ f_N(q) = \sum_{n=1}^\infty a_n q^n $$ to $C_N$. If the Pell conic is modular, then $f_N$ is a rational function of $q$ (the numerators are basically what are known as Fekete polynomials, and a partial fraction decomposition naturally will lead you to Gauss sums) and can be extended to a meromorphic function on the whole complex plane with poles at the primitive $N$-th roots of unity and satisfying a functional equation $$f_N\Big( \frac1q \Big) = - \chi(-1) f_N(q). $$ Observe that if $\chi(-1) = 1$, then $f_N(1) = 0$. This is closely related to the global solvability of the Pell equation in integers. If $\chi(-1) = -1$, on the other hand, then $f_N(1) = 2h/w$, where $h$ is the class number and $w$ the number of roots of unity in the complex quadratic number field with discriminant $-N$.

4. For a prime $p \nmid N$, define the Hecke operator $T_p$ by $$ f|_{T_p}(z) = \frac1p \sum_{a=0}^{p-1} f\Big(\frac{z+a}p \Big). $$ Then the Pell forms $f_N$ are simultaneous eigenforms of the $T_p$ with eigenvalues $a_p$.

   There are other eigenfunctions of $T_p$: the cotangent function, the Bernoulli polynomials, Hurwitz's zeta function etc. This is closely related to what Kubert and Lang have called distributions (cf. etc. Washington's cyclotomic fields).

5. The modular analog of Pell conics are the cyclotomic units. You can parametrize Pell conics analytically by trigonometric functions and "arithmetically" by taking norms from cyclotomic units.

Edit. Here's the link to the preprint.

Answer 3

Let me give at least a partial answer.

If you want to view the 1 and 2 dimensional theories uniformly, you should look at everything adelically. In dimension 1, Dirichlet characters can be viewed as idele class (or Hecke) characters, which is to say irreducible representations of $\mathbb Q^\times \backslash \mathbb A^\times$. Hence 1-dimensional Galois representations correspond to automorphic representations of $GL_1(\mathbb A)$. (Edit: I forgot to mention, this correspondence is abelian class field theory!)

In general, $n$-dimensional Galois representations conjecturally correspond to automorphic representations of $GL_n(\mathbb A)$. There's a standard passage from modular forms to (classical then adelic) automorphic forms, and from automorphic forms to automorphic representations of $GL_2(\mathbb A)$. The "odd" 2-dimensional Galois representations should correspond to automorphic representations coming from modular forms. (Incidentally, to get the Euler product for the L-function of a modular form, it should be an eigenform.) Interpreted in terms of automorphic representations, one sees that (the L-functions for) modular forms are a higher dimensional analogue of (the L-functions for) Dirichlet characters.

To change the base field, work with representations of $Gal(\bar K/K)$ on the Galois side, and $GL_n(\mathbb A_K)$ on the automorphic side.

For the last question, if $K/\mathbb Q$ is Galois, then yes, you have the same type of factorization for the Dedekind zeta function (realize it as an Artin L-function for the trivial representation).

There are several places you can read survey articles. For instance, the book "Introduction to the Langlands Program." Knapp also has nice survey articles, and Gelbart has a nice Bulletin article. I have links to some articles here.