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This seems like something that should be well-known, but as an outsider to the field, I'm having trouble locating precise statements.

Hasse-Weil zeta functions of Shimura varieties should be alternating products of automorphic $L$-functions. This seems to be known when the underlying group is $GL_2$ (or a quaternion division algebra) over a totally real field, $GSp_4$ over $\mathbb Q$ (maybe a totally real field), unitary groups in three variables over $\mathbb Q$ (maybe a totally real field), and maybe certain other unitary groups.

In "Where Stands Functoriality Today?", Langlands writes that the "principal factors" of the zeta function of (general) Siegel modular varieties are automorphic $L$-functions attached to spinor representations (which have not been analytically continued), so some more general calculations seem to be known...

My question: For which groups are the Hasse-Weil zeta functions of the associated Shimura varieties known to be alternating products of automorphic $L$-functions (maybe modulo "technical" restrictions and bad prime calculations)?

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This question is perhaps answered in the first few paragraphs of the following paper of Milne on the subject: jmilne.org/math/articles/2008b.pdf –  Kevin Buzzard Sep 16 '11 at 19:50
Kevin, that would be nice! I don't quite see it (though as an outsider, I may miss certain implications). Milne says that to show that the zeta function of a Shimura variety is automorphic, you need 1) a description of the points of the variety over a finite field 2) "a combinatorial argument" using the trace formula and the fundamental lemma. Milne's paper (and his references) just deals with 1). I don't think 2) comes for free (right?). But it is nice to see that there are some somewhat general results known for groups of type A and C. –  B R Sep 16 '11 at 20:44
I think the fundamental lemma comes for free nowadays. As for the current state of stabilizing the trace formula I'd be tempted to look at Harris' book project: fa.institut.math.jussieu.fr/node/29 . I'm sorry I can't just tell you the answer -- I am no expert. –  Kevin Buzzard Sep 17 '11 at 10:47
Thanks for the links, they did give me some ideas for searching the literature. –  B R Sep 17 '11 at 18:57
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1 Answer

up vote 12 down vote accepted

As far as I understand, what you need to do to prove that the zeta function of a Shimura variety is automorphic is (I'm ignoring the bad primes here, but I think that we now know enough about them too - I can develop later if you want) :

(1) Do some kind of point-counting over finite fields. For PEL Shimura varieties of type A and C, Kottwitz has done it. Actually it's a bit more, you calculate the trace of a Hecke operator times a power of the Frobenius (at a place of good reduction) on the cohomology with compact support.

(2) "Stabilize" the resulting formula so you can compare it with Arthur's stable trace formula. Here there is a choice. As the reason we expect the trace of a Hecke operator to compare well to the trace formula is Matsushima's formula, as Matsushima's formula for noncompact Shimura varieties (due to Borel and Casselman in that case) is a formula for the L^2 cohomology of the Shimura variety, and as the algebraic avatar of this L^2 cohomology is the intersection cohomology of the minimal compactification, you can choose to first extend the result of (1) to this intersection cohomology, and then "stabilize" whatever you get. Or you can ignore this and choose to work with compact support cohomology, at the cost of greater complication on the trace formula side. Laumon followed that approach for Siegel modular threefolds, but as far as I know, in other cases people generally choose the first approach. Anyway, this is not a problem for compact Shimura varieties. For compact PEL Shimura varieties of type A and C, this "stabilization" part is also due to Kottwitz (in the first part of his beautiful Ann Arbor article "Shimura varieties and $\lambda$-adic representations"), we also know some noncompact PEL cases of type A and C by Morel's work, and basically all other type A and C PEL cases should reduce to Kottwitz's calculations, but I don't think this is written anywhere.

Oh, and you need the fundamental lemma for this part.

(3) Now you still need to compare the result of (2) with the stable trace formula, so obviously you need to know the stable trace formula (here you need the weighted fundamental lemma), and to make sense of the results and get a nice formula for your zeta functions you also want to know Arthur's conjectures on the classification of discrete automorphic representations. How you get the zeta function formula if you assume Arthur's conjectures is explained in the second part of Kottwitz's previously-cited beautiful Ann Arbor article.

So the question is, what do we know about Arthur's conjectures ? Well, they are accessible. Arthur is supposed to be writing a proof in the case of symplectic groups, and he is actually making progress on it. Note however that for Siegel modular varieties, you'll need general symplectic groups, so there will be a further reduction step even after Arthur finishes writing his book. (But we are nearing a proof of the automorphy of the zeta function. Yay !) There are a few young and brave ones who are planning to tackle the case of unitary groups (if I remember well, Sug Woo Shin, Tasho Kaletha, Paul-James White and Alberto Minguez). A last word of caution, all this (Arthur's work and the others' future work) depends on the stabilization of the twisted trace formula, which is at the moment not totally written down, I'm afraid, but there's a group of Serious People in Paris (like Clozel, Waldspurger etc) who have vowed to take care of it.

So, to sum up, It's Complicated, but we seem to be close for PEL Shimura varieties of type A and C, especially Siegel modular varieties. Also, if you just want the zeta function to be a product of automorphic L-functions with complex exponents (and not integral exponents), then I think that's known for PEL case A (maybe not written in all cases, though); whether you can make these exponents rational without too much additional work, I am not sure (it seems that this should be an easier thing than proving they're integers).

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Thanks! I want to clarify something. When you write "Arthur is supposed to be writing a proof in the case of symplectic groups [...] you'll need general symplectic groups, so there will be a further reduction step", do you mean that Arthur is working on Sp(2n), and we need results for GSp(2n) (or something else)? Also, for completeness, nothing is known for Types B and D (or E_6 and E_7) (though the trace-formulaic aspects seem to be no more difficult than the other classical groups (at least for B and D), so the problem is contained in counting points), correct? –  B R Sep 17 '11 at 19:38
Yes, I did mean that Arthur is working on Sp(2n) and we want GSp(2n) (or PSp(2n)) for the application to Shimura varieties. As for the other type : I don't know anything about exceptional types, and there has been some progress on non-PEL classical type Shimura varieties, but as far as I know we still don't have a counting point formula. Arthur is also working on his conjecture for orthogonal groups (you can't really separate them from symplectic groups, as they appear as endoscopic groups anyway), and his articles on the trace formula are written in the genetal setting. –  Alex Sep 18 '11 at 15:20
However, a lot Kottwitz's papers (including the ones I cited and the second half of the counting point article) make the assumption that the group has a simply connected derived group (the reason for that assumption is that it ensures that every semisimple element will have a connected centralizer, and so it makes everything simpler as centralizers appear all the time in integral orbital, parametrizing stable conjugacy classes etc). This excludes orthogonal groups. So I would not say that the problem is contained in counting points. There is some more work to be done beside that. –  Alex Sep 18 '11 at 15:24
Also, I forgot to mention that at least one of Kottwitz's papers on the subject is unpublished. And finally, this is something I probably should have written in my answer : The program I outlined allows you to prove the automorphy of the zeta function of the intersection cohomology of the minimal compactification. If you want the zeta function of the Shimura variety itself, you need a fourth step that is a formula giving the cohomology with compact support of the Shimura variety as the intersection cohomology of its minimal compactification plus an alternating sum of... –  Alex Sep 18 '11 at 15:27
...intersection cohomology of minimal compactifications of smaller-dimensional Shimura varieties (maybe with coefficients in non-trivial local systems). Fortunately, such a formula is known. This would give the automorphy of the zeta function of the Shimura variety, but I don't know if there is a nice conjectural description of excatly which automorphic representations appear in the compact support cohomology (as there is for the intersection cohomology, which was the reason that it is easier to take a detour through it). –  Alex Sep 18 '11 at 15:30
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