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Apologies in advance if this is a bit too simple to ask here, but I think I'm probably more likely to get an answer here than at stackexchange.

I've been trying to learn the basics of the Langlands program over the last couple of months, and I've reached a funny point where I think I understand the rough idea of whats going on, but I seem to have a fairly large amount of reading to do before I can take the next step and understand everything properly.

I'm currently at the point where I'm completely happy with Tate's thesis and the basic ideas of automorphic forms and representations. My understanding of Langlands right now is roughly:

  • Tate's thesis shows us that by using abelian harmonic analysis on the adele ring, we can prove functional equations for L-functions attached to Hecke characters.

  • Hecke characters are just one-dimensional automorphic representations.

  • We can (or believe we can? I'm not entirely sure what the status of this part is) use nonabelian harmonic analysis to prove functional equations for the L-functions attached to more general automorphic representations. I know the point of the fundamental lemma is that it lets us use the information given by the Arthur-Selberg trace formula in a useful way - so this generalises the role of Poisson summation in Tate's thesis?

  • A (very) rough statement of the local Langlands conjectures is that a whole load of arithmetic L-functions that we've defined actually arise from automorphic representations.

  • So local Langlands would mean we can prove functional equations for a huge amount of L-functions.

I'm aware that I've missed out huge chunks of the program like functoriality, which I haven't even begun to understand yet, but I'd like to think there's enough there for me to have a pretty reasonable idea of what's going on at a basic level.

Then my question is this: is this at all accurate? The idea as I've summarised it above is one that I've pieced together myself from the various things I've read about Langlands, and it seems like a pretty simple explanation of why we should care about Langlands, yet it's not an explanation I can recall seeing anywhere before, which makes me suspect that I might have missed the point a bit.

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Local Langlands is a local story; automorphic representations are global objects. A very rough statement of the global Langlands conjectures is that a whole load of arithmetic $L$-functions arise from automorphic representations. If by "arithmetic $L$-function" you mean "$L$-function attached to a motive" or, more mundanely, "to a Galois representation", then functoriality is just the statement that there is a construction on the automorphic side which matches goofing around on the Galois side, e.g. taking the symmetric square of the auto rep. –  user30035 Jan 19 '13 at 16:14

2 Answers 2

up vote 8 down vote accepted

I would disagree with your last two points just as wccanard does in his comment: automorphicity of $L$-functions is part of global Langlands functoriality, not the local conjectures (although the two are related).

I would also disagree with your third point: instead of nonabelian harmonic analysis, it is automorphicity that translates into functional equations and vice versa. For example, by the automorphicity of certain Eisenstein series we can see that certain $L$-functions coming from automorphic forms satisfy a functional equation, and from this we can sometimes deduce that the $L$-function is itself automorphic (not just pretends to be).

Regarding nonabelian harmonic analysis I would say that it naturally leads to the notion of automorphic forms and representations (via the spectral decomposition), and it also provides a good framework to study them (including establishing certain cases of functoriality without using $L$-functions).

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Ah okay, so I had really missed the point a bit. Thanks for clearing that up! –  user19918273 Jan 20 '13 at 15:37

To complement the answer from GH from MO:

Your first two bullet points, about Tate's thesis and Hecke characters are completely correct.

Your third point should be made more precise as follow. Yes we can prove the functional equation for the L-function attached to automorphic representations of $GL_n$, and a bunch of other things for them: that they have a meromorphic continuation, where the poles are if any, then as you say the functional equation, and then the non-vanishing of these L-function on the line $\Re s=1$ (the analog of the Hadamard-De la Vallée Poussin theorem). Roughly we know these automorphic $L$-functions almost as well (or as badly) as the Riemann Zeta function. ("Almost" because we don't know the Ramanujan hypothesis, that is we only know that the series defining these functions converges in a half-plane slightly smaller as what we expect, but in the grand scheme of things that's a detail. "as badly", because like for the Riemann Zeta Function, we expect them to satisfy the Riemann hypothesis, and we have no idea how to prove it.) All these results are due to Jacquet-Shalika (except for $n=1$, where they are due to Hecke and Tate), and were proved relatively early in the development of the theory of automorphic forms.And while Jacquet-Shalika's proof are hard and very clever, they don't use the Arthur-Selberg trace formula, let alone the fundamental lemma which was proved much later.

For your fourth point, as GH says, it would be correct (as a very rough statement) if you replaced "local" by "global". And the same is true for your fifth.

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