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Suppose $\pi$ is an irreducible automorphic representation of a reductive connected algebraic group $G$ over $\mathbb{A}_K$, here $K$ is a number field, $\mathbb{A}_K$ denotes its adeles. We have a restricted tensor product decomposition of $\pi=\otimes\pi_v$, where $\pi_v$ is an irreducible admissible representation for $G(K_v)$, and for all but finitely many $v$, $\pi_v$ is unramified.

We know how to define local L-factors at $v$ is $\pi_v$ is unramified, and we also know how to define local L-factors at archimedean places because of Langlands classification. So the question is how to define L-factors at ramified places?

As far as I know, at least for $GL_n$, we can define it as the gcd of some family of integrals via integral representation of L-function.

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3 Answers 3

I think you're slightly misled. $\pi$ doesn't have an $L$-function "in abstracta". An unramified $\pi_v$ at a finite place $v$ gives rise, by Langlands' interpretation of the Satake isomorphism, to a semi-simple conjugacy class in the local $L$-group (or, more fancily, to an unramified representation of the local Weil[-Deligne] group [with N=0]), and that's not enough for an Euler factor. So what one does is also fixes a representation $r:{}^LG\to GL_n(\mathbf{C})$. Now one has an $L$-function $L(\pi,r,s)$, it depends on both $\pi$ and $r$ though. For example, if you choose a modular elliptic curve over the rationals, but let $r$ be the bazillion'th tensor power of the standard 2-dimensional representation of $GL_2$, you have an $L$-function which no-one knows how to analytically continue.

But the real answer to your question is that this is an open problem. One would like to say that by functoriality there is a representation $r_*(\pi)$ on $GL_n$ and you use standard definitions of $L$-functions on $GL_n$. However the existence of $r_*(\pi)$ is a fundamental open problem: Langlands functoriality. Defining local $L$-functions at the ramified places is a tiny tiny piece of this open problem, but as far as I know it's also open.

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I think that "bazillion'th" and "tiny tiny" are funny exaggerations. I think that once you get past the symmetric 4th power L-function for GL(2), not too much is known, though one gets meromorphic continuation somewhere near the symmetric 10th, I think, using a combination of methods. I think that a bazillion is definitely bigger than 10, probably bigger than 20 too. Also, defining local L-functions at the ramified places is a pretty huge (not tiny tiny) piece of Langlands functoriality. By converse theorems, local L-functions and epsilon-factors characterize functoriality! –  Marty Mar 28 '10 at 11:26
    
@Marty: this style actually represents laziness on my part. I know that much is known about symmetric squares, a fair bit is known about symmetric cubes and so on. But I do not know where our knowledge peters out. So rather than writing "Symmetric 5th power" and risking being wrong because of some recent preprint on ArXiv I write "bazillionth power" on the basis that if someone comes along and points me to some statements about 6th powers I can happily assure them that a bazillion is much bigger than 6. As for L-functions: you need more than a local definition: you need an ctn too as you know –  Kevin Buzzard Mar 28 '10 at 11:52
    
@Kevin: My comment wasn't a criticism -- just a refinement. As for converse theorems and L-functions and epsilon-factors, I was referring to "Local Converse Theorems". By results of Henniart, and later results of J. Chen, the epsilon factors of twists are enough to characterize irreps of GL_n over a p-adic field. So knowledge of L- and epsilon- factors $L(\pi, r,s)$ would suffice to "nail down" functoriality purely locally. Here I require "lots" of L- and epsilon- factors, corresponding to lots of "twists" of $\pi$ by irreps of $GL_m$ (with $m \leq n-1$ or $n-2$ maybe). –  Marty Mar 28 '10 at 18:04
    
@Marty: aah yes, I misunderstood what you were saying about $L$-functions. I always thought a converse theorem was a global statement. So the last sentence of my previous comment needs deleting, but we can't edit comments :-( –  Kevin Buzzard Mar 28 '10 at 20:16
    
Actually, I guess then the emphasis of the comment moves: if you have enough local L-functions then perhaps you'll get local Langlands, but this is still a long way from global functoriality (which is what I thought we were talking about ;-) but perhaps you weren't!) –  Kevin Buzzard Mar 28 '10 at 20:18

This question was posted a while back but I just saw it. Here are some thoughts. In practice there are a couple of methods to construct L functions for local ramified representations. The first one is the Langlands-Shahidi method which works for "generic representations" of quasi-split groups and the second one is the method of integral representations (Rankin-Selberg method, Shimura's integral and the doubling method, etc). It is probably a bit painful to give a meaningful description of these two methodologies in such a limited space, so instead let me refer you to a couple of places where you can see accessible accounts of the two approaches. A good reference for the basics of the Langlands-Shahidi method is the beautiful monograph "Analytic properties of automorphic L functions" by Gelbart and Shahidi. The same reference has a nice introduction to the method of integral representations. Dan Bump has written two very informative survey papers on the Rankin-Selberg method. Cogdell's ICTP lectures on the Rankin-Selberg method are lovely. Another good book to look at is the AMS book by Cogdell, Kim, Murty.

As it stands there is no Langlands-Shahidi method for non-generic representations. What is missing from the picture is a good supply of easy to use unique models, like the Whittaker model in the generic setting. For orthogonal groups, however, recent progress by Waldspurger and others on the Gross-Prasad conjectures gives one the hope that maybe one can now develop a Langlands-Shahidi method, although there are serious obstacles to deal with.

Most of the integral representations known to mankind too are closely linked with unique models (Whittaker, Bessel, etc). Sakellaridis has a theory that "explains" (some) integral representations in terms of spherical subgroups of reductive groups.

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As far as I understand, attaching an $L$-function to an automorphic representation attached to a general reductive group $G$ is conjectural and still open.

The way one attaches $L$-function depends on a representation $r$ of ${^L}G$ and partitioning the set of irreducible admissible representation of $G(k_v)$ in to $L$-packets (which is conjectural in general and known in very few cases). Assuming one can define local $L$-packets, Borel and Tate's article in Corvalis explains how to attach $L$-function to it. But still this $L$-function depends on the chosen representation $r$.

If $\pi$ is an irreducible admissible representation of $G_A$ then $\pi= \otimes_v \pi_v$, where $\pi_v$ is an irreducible admissible representation of $G(k_v)$. So assuming we can partition the set of irreducible admissible representation of $G(k_v)$ in to $L$-packets, $\pi_v$ belongs to $L$-packet $\Pi_{\phi_v}$ corresponding to some admissible homomorphism $\phi_v$ of Weil-Deligne group to ${^L}G/k_v$ . The repesenation $r$ defines a representation $r_v$ of ${^L}G/k_v$.

Then the $L$-function attached to $\pi$ and $r$ is defined as: $L(s,\pi,r) = \prod_v L(s,\pi_v,r_v)$, $L(s,\pi_v,r_v)=L(s, r_v \circ \phi_v)$

Now $r_v \circ \phi_v$ is a represenatation Wiel-Deilgne group, so by Tate's article in Corvalis, we know local $L$-factor.

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