# L-functions and higher-dimensional Eichler-Shimura relation

From what I have been reading I understand that it is a part of the Langlands program to express Zeta-function of a Shimura variety associated to the group G in terms of L-functions attached to G. I am not sure to what extent this already has been proven.

In the cases I "know", which are the modular curves and Shimura curves - the essential step is to use the Eichler-Shimura congruence relation, which gives a connection between the Frobenius action and the Hecke correspondences.

Now, in the higher-dimensional case, say that of Shimura varieties of PEL-type, is there a generalization of the Eichler-Shimura relation, and does this generalization play the same role in the proof of the statement?

Maybe someone could give a rough overview of what is used in the proof in the known cases (as far as I understand, PEL case is known)?

Thank you.

-
"... in terms of L-functions attached to G." and maybe some other groups if the variety is not compact. While waiting for answers from experts, I'd like to bring up R. Taylor's article on Siegel 3-folds (around 1993 I guess) generalizing the congruence relation. –  shenghao Dec 20 '10 at 23:26

Surprisingly, the case of modular curves is misleading! General theory of correspondences, plus the theory of the mod $p$ reduction of curves like $X_0(Np)$ ($p$ doesn't divide $N$) give a relationship between the Hecke operator $T_p$ and the Frobenius endomorphism $Frob_p$ acting on, say, the Jacobian of a modular curve. But now there's a coincidence: from this data we can figure out a degree two polynomial satisfied by $Frob_p$ on the chunk of the Tate module of the Jacobian which corresponds to a given weight 2 eigenform, and we know for other reasons that this chunk is 2-dimensional, and it's not hard to tease out from this a proof that the degree two poly we know must be the char poly of Frobenius. I would recommend Brian Conrad's appendix to the Ribet-Stein paper for a modern treatment of this story.

But pretty much the moment one leaves the relative safety of curves, this trick doesn't work, because the arithmetic geometry gives you a polynomial which kills Frobenius, but you don't have enough information to check that this poly is the char poly. So you have to do something vastly more complicated! This vastly more complicated thing was figured out by Langlands in the case of modular curves (the details are mostly buried in his Antwerp paper). Another place to look for the modular curve case is Clozel's Bourbaki article from about 1993, although he skips many of the details when it comes to cusps and supersingular points.

OK so Langlands' proof looks like this. We need to check that the $L$-function of a modular curve is a product of $L$-functions coming from automorphic forms. The $L$-function of a modular curve is a product of local terms, and the local term at $p$ can be computed if one knows the number of $k$-points on the curve for $k$ running through all finite fields of char $p$. The goal is to relate these numbers to numbers coming from the theory of automorphic forms---basically numbers related to the Hecke operators at $p$. Langlands observes however that he can count the number of $k$-points on a modular curve: there are cusps, which aren't too hard, and then there are supersingular points, which come out as some explicit number, and then there are ordinary points, which he counts isogeny class by isogeny class. Langlands can compute isogeny classes using Honda-Tate theory and then he writes a formula for the size of each isogeny class, via some manipulations with lattices, as an integral over some adele group (the point is that you use the Tate module of the curve at $\ell$ to understand $\ell$-power isogenies, and then rewrite such things as integrals over $\ell$-adic groups, and then take the product over all $\ell$).

The resulting equation is a horrific mess, which can basically be almost completely written in terms of twisted orbital integrals on $GL(2)$ and various subgroups. The reason the orbital integrals are twisted is that the arguments for isogenies at $p$ use the Dieudonne module of the curve, so one has to do semilinear algebra here to count subgroups of $p$-power order.

Next Langlands uses the Fundamental Lemma for $GL(2)$, which he can prove with his bare hands, to turn his twisted orbital integrals into orbital integrals.

Finally, Langlands looks at the resulting mess and says "oh look, this is just precisely the geometric side of the trace formula! So it equals the spectral side.". And then he looks to see what he has proved, and he's proved that the size of $X_0(N)(k)$ equals the trace of some Hecke operator (depending on $k$) and he figures out how this trace relates to $T_p$ and then he's proved Eichler-Shimura.

I know of no complete reference for the above argument. In particular, checking that the "nasty" terms in the trace formula coming from the fact that we're in the non-compact case, match up precisely with the number of cusps, is something so deeply embedded in Langlands Antwerp that I have never really managed to get it out.

This method is much more complicated than Eichler-Shimura, but it generalises much better. By Corvallis they basically had got it working for forms of $GL(2)$ over a totally real field, where already the arithmetic geometry methods were struggling. See the articles of Kottwitz, Milne, Casselman etc.

Later on Kottwitz pushed the method through for a wide class of PEL Shimura varieties, but I'm not sure he did it for all of them. This is what prompted Clozel's Sem Bourbaki talk in 1993 or so---I mentioned this earlier. I think Milne has written a lot about this area recently though, in particular I believe he's trying to debunk the myth that the result is proved for all PEL Shimura varieties. See in particular Milne's paper "Points on Shimura varieties over finite fields: the conjecture of Langlands and Rapoport" (2008) available here for example.

-
"By Corvallis ... ": as far as I understand, these articles in Corvalis are a report on Langlands's Canadian Bulletin article, which already goes further than Corvalis (e.g. it treats some cases of endoscopy). Note also that there is a generalization of the Eichler--Shimura relation, it has been proved in many PEL cases (by Wedhorn, and maybe others too --- e.g. in the symplectic case I think it was proved by Faltings and Chai), and one can sometimes use it to extract information about the Galois action on etale cohomology (as you well know, e.g. your advisor did this for $GSp_4$). –  Emerton Dec 21 '10 at 1:12
A good reference for the method Kevin is talking about seems to be the paper of Scholze arxiv.org/pdf/1003.1935.pdf In this paper he does rather more (also checking the equivalent of an Eichler Shimura relation at bad places), but it isn't too difficult to fish out the pieces you want, in particular he checks the boundary components line up. –  Tom Lovering Apr 6 '13 at 19:16