Which Shimura varieties are known to be automorphic? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:35:28Z http://mathoverflow.net/feeds/question/75632 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75632/which-shimura-varieties-are-known-to-be-automorphic Which Shimura varieties are known to be automorphic? BR 2011-09-16T19:12:40Z 2011-09-17T16:57:29Z <p>This seems like something that should be well-known, but as an outsider to the field, I'm having trouble locating precise statements.</p> <p>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. </p> <p>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...</p> <p>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)?</p> http://mathoverflow.net/questions/75632/which-shimura-varieties-are-known-to-be-automorphic/75684#75684 Answer by Alex for Which Shimura varieties are known to be automorphic? Alex 2011-09-17T16:57:29Z 2011-09-17T16:57:29Z <p>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) :</p> <p>(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.</p> <p>(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.</p> <p>Oh, and you need the fundamental lemma for this part.</p> <p>(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. </p> <p>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.</p> <p>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).</p>