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There are Artin $L$-series and Dirichlet $L$-series, and zeta functions for varieties and for number fields; there are a slew of objects named after Hecke... There are also various kinds of characters in these areas.

I've always had a hard time keeping all of them straight. I would be very grateful if someone could put them in some sort of perspective that would make it clear what the role of each of those objects is with respect to the rest.

If you need to use the language of Langlands, that would be acceptable. (perhaps it is best to see it in those terms?)

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My naive understanding is this: some of these series are the ones we "really" care about (the ones attached to number fields, varieties, etc.) and the others are the ones with nice analytic properties (the ones attached to characters, modular forms, etc.), and we want to express the former in terms of the latter so that the nice properties of the latter can be used to study the former. –  Qiaochu Yuan Dec 14 '10 at 5:26
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Dear Qiaochu, I dare you to express the sentiment of your comment with an analytic number theorist in the audience! (Written with a smile; my own work definitely accords with the sentiment of your comment. But if you are trying to count primes, then the $\zeta$-function and Dirichlet $L$-functions are the most important objects, and you don't need to care about things on the motivic side. For other problems, it is the spectrum of the Laplacian, say on $L^2(\mathcal H/SL_2(\mathbb Z))$ that matters, and then you are focussing on Maass forms --- so which ones you "really" care about ... –  Matt E Dec 14 '10 at 5:40
    
... depends on what kind of problems you are investigating.) –  Matt E Dec 14 '10 at 5:45
    
@Matt E: yes, that's why I put "really" in quotes :) –  Qiaochu Yuan Dec 14 '10 at 6:08
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By the way, one would normally call these Dirichlet series rather than power series. –  Matt E Dec 14 '10 at 6:43
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1 Answer

up vote 13 down vote accepted

First of all, one shouldn't distinguish too much between the terms $L$-series and $\zeta$-function; it is more or less a matter of history which term you use in a given context.

The true distinction is between objects on the "motivic" (Diophantine equation or algebraic number theory) side, and objects on the "automorphic" side.

So, Artin $L$-functions, and $\zeta$-functions and $L$-functions attached to varieties, are on the motivic side. They don't have obvious analytic continuations or functional equations.

Dirichlet $L$-functions (i.e. the ones built out of characters of the multiplicative group mod $N$ for some $N$), Hecke $L$-functions (built out of ideal class group characters), $L$-functions of modular forms, or more generally automorphic forms, are automorphic $L$-functions. Except for the last (general automorphic $L$-functions) these can be proved to have analytic continuation with functional equations. The natural limit of these analytic continuation/functional equation arguments (which begin with Riemann, with Tate's thesis being another major point en route) is the fact that standard $L$-functions for automorphic forms on $GL_n$ have analytic continuation/functional equations.

Langlands's functoriality conjecture in particular conjectures that any (a priori more general) automorphic $L$-function is in fact a standard $L$-function. Many special cases are known (and Ngo got the Fields medal for proving the fundamental lemma, which is one tool in proving certain cases, namely those arising from endoscopy), but it is wide open in general.

The reciprocity conjecture then states that any of the motivic $L$-functions are also actually standard $L$-functions. (E.g. for $L$-functions of elliptic curves over $\mathbb Q$, this becomes the statement that any such elliptic curve is associated to a weight 2 modular form, which was proved by Wiles et. al.) This is also wide open in general.

The special case of Artin $L$-functions, and some other special cases, are actually covered by both conjectures. (This is because, if you broaden your perspective sufficiently, as Langlands did, by allowing arbitrary reductive algebraic groups into the picture, then Artin $L$-functions can be thought of as a particular kind of automorphic $L$-functions. This is not so easy to see when you are just entering the subject, but one can think of just the Riemann $\zeta$-function: this is certainly motivic, being the $\zeta$-function of the variety Spec $\mathbb Q$, but is also automorphic, being the Dirichlet $L$-function for the trivial character.)

But in general, the conjectures have a slightly different nature: functoriality can be thought of (if you want) purely in terms of harmonic analysis on adelic groups, whereas by its very nature reciprocity involves arithmetic geometry. In practice, at least at the moment, the two seem to be fairly intertwined. Functoriality certainly is a tool that can be very helpful in proving reciprocity; also, of those cases of the Artin conjecture which are known, some are proved using the functoriality view-point (e.g. Langlands--Tunnell) and some using the reciprocity view-point (e.g. Buzzard--Dickinson--Shepherd-Barron--Taylor).

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