# The monodromy-weight-, Ramanujan-, Langlands-landscape

The drawing on the last page of Yoshida's notes make me puzzle, perhaps you can help? It shows a "landscape" featuring the monodromy-weight conj., the general Ramanujan-conj., the Langlands correspondence and some connections between them. However, I guess the Ramanujan-conj. "is" the monodromy-weight-conj. expressed via Langlands correpondence and the Rapoport-Zink spectral sequence is a tool for showing the MWC in some cases and reducing the computation of L-functions to semi-simple L-functions. That seems not to fit to the drawing. Can you can clearify that?

Edit: A fascinating new survey on the "philosophy of weights" with many conjectures by Jannses. (Posted here because I don't know an entry where it would fit better)

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This page looks extremely interesting, but don't you think it's a little too vague and hurried for anyone to come up with an answer to your question? It's on one page of a talk which begins with Fermat's theorem on $x^2+y^2$, after all. Why not just write to Yoshida directly? When everything's clear, please answer your own question here on MO! –  Jared Weinstein Dec 9 '09 at 18:07
To me, it looks less like a landscape and more like a game of Donkey Kong. Presumably there is a program for proving something interesting encoded in there, but without additional context, I can't say anything precise. –  S. Carnahan Dec 10 '09 at 5:39

Dear Thomas,

This "landscape" is, I think, a sketch of the proof of the following theorem of Taylor and Yoshida: if $\Pi$ is a self-dual cuspdidal automorphic form on $GL_n$ over $E$ (a CM field) (satisfying some further technical conditions) and $\rho$ is the associated $n$-dimensional Galois representation (constructed by Harris and Taylor in their book "The geometry and cohomology of some simple Shimura varieties"), then the local factors of $\Pi$ at any prime $p$ of $E$ matches, via local Langlands, with the restriction of $\rho$ to a decomposition group at $p$.

What was proved in Harris--Taylor was that this matching is correct, up to the question of matching the $N$ on each side. What Taylor and Yoshida verified is that the $N$ on each side also matches.

The top part of the landscape represents the reduction to the unipotent situation (i.e. the context in which $\Pi$, locally at $p$, has an Iwahori fixed vector --- the analogue of classical level $\Gamma_0(p)$), by base-changing to an extension of $E$. On the automorphic side, this is the process labelled by the names Arthur--Clozel--Selberg; on the Galois side, this is just restricting from $Gal(\overline{E}/E)$ to $Gal(\overline{E}/E')$ for some appropriately chosen extension $E'$ of $E$. If one thinks of $\rho$ as appearing in the cohomology of a Shimura variety (which is, after all, how it is constructed), then this just corresponds to base-changing the variety from $E$ to $E'$, which is why this process is labelled as geometric base change in the "landscape".

On the lower right of the diagram, one has the generalized Ramanujan conjecture, stating that the local factor of $\Pi$ at $p$ should be tempered, which in turn implies that the local (at $p$) Galois representation (more precisely, Weil--Deligne representation) attached to this local factor satisfies the monodromy weight conjecture. This temperedness was proved by Harris--Taylor.

Now we already know that $\rho$ locally at $p$ matches with the local factor of $\Pi$ at $p$ up to $N$, and it is not hard to show that there is a unique way to add $N$ so as to obtain a local Weil--Deligne representation satisfying the monodromy weight conjecture. So to prove their theorem, Taylor and Yoshida are reduced to proving that $\rho$, locally at $p$, satisfies the monodromy weight conjecture.
They do this via an application of the Rapoport--Zink spectral sequence and a careful analysis of the bad reduction of the Shimura variety in whose cohomology $\rho$ lives. This is represented in the lower left part of the diagram.

The very bottom part of the diagram on the left represents the fact that a priori one knows that $\rho$ is mixed, i.e. its semi-simplification has Frobenius eigenvalues that are Weil numbers of various weights. But the main part of the landscape here is the monodromy weight conjecture, which describes the precise relationship between the Frobenius eigenvalues and the $N$ operator.

To prove the MWC in their context, Taylor and Yoshida use the interplay between the geometry of the special fibre of the Shimura variety and the representation theory of $\Pi$ that is the main subject of Harris and Taylor's book, as well as Section 2 of Taylor and Yoshida's article. In particular, the fact that $\Pi$, locally at $p$, is tempered unitary, and is assumed to have an Iwahori fixed vector, puts strong restrictions on its structure (using the known classification of unitary reps. of $GL_n$ over local fields; this is why Tadic's name appears on the right hand side of the landscape), which when fed over to the geometric side implies that the RZ spectral sequence degenerates at $E_1$, giving the desired monodromy weight conjecture.

My suggestion is that if you want to understand this in more detail, you should read Taylor and Yoshida's article. Start at the very end of the paper, where the main theorem is proved (i.e. the paragraph beginning "We can now conclude ... "). The lemmas referred to in section 1 are more or less elementary. Theorem 3.2 is the heart of the argument; it is where degeneration of the RZ spectral sequence is proved. One can try to read it more or less formally, taking various assertions as a black box, at least to see what role the temperedness of $\Pi$ locally at $p$ plays. The earlier parts of Section 3 are just establishing notation, and making contact with the book of Harris--Taylor. Section 2 is devoted to establishing the basic properties of the semi-stable models of the Shimura varieties in whose cohomology $\rho$ lives; I recommend treating it as a black box on first reading.

Added: First, I should note that the references I make are to the version of Taylor--Yoshida that is currently posted on Taylor's web-page, which I am told may differ quite substantially in its organization from the published version of the paper.

Also, let me add something about the role of the classification of unitary representations of $GL_n$, which will help illuminate the structure of the lower part of the landscape: First, one uses the general result that geometrically obtained Galois representations are mixed (i.e. have Frobenius eigenvalues that are Weil numbers), coupled with the local-global compatibility without the $N$, to deduce that the Frobenius eigenvalues of the Weil--Deligne representation attached to the local factor of $\Pi$ at $p$ are Weil numbers. Second, results of Tadic on the classification of unitary representations of $GL_n$ of $p$-adic fields greatly restrict the structure of this local factor (since it is both unitary and generic, being the local factor of a cuspidal automorphic representation for $GL_n$). Finally, when this is combined with the fact that the Weil--Deligne rep'n associated to it via local Langlands is mixed, ones sees that this local factor is forced to be tempered. This is how Harris--Taylor deduce temperedness; it is an analogue at $p$ of an argument at the archimedean prime made by Clozel in his Ann Arbor paper. (See Lemma 4.9 on p.144 of volume I of the Ann Arbor conference. A scan is available on Jim Milne's web-page here, but note that it is quite a big file.)

Finally, let me now explain how to correctly read the landscape: one starts in the lower left, following the arrow. When you come to a bridge, you cross the river from the motivic/Galois side to the automorphic side, or back again, continuing to follow the arrows. The areas marked with x's are impenetrable (or, at least, you don't try to cross them directly --- e.g. you don't prove GRC directly, you don't prove WMC directly, you don't study the general semi-stable reduction problem directly); the only way to proceed is by crossing the river. This then gives the structure of the Taylor--Yoshida argument.

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Dear Matthew, many thanks! –  Thomas Riepe Aug 28 '10 at 7:25
What a wonderful answer! Many thanks, indeed. –  Keerthi Madapusi Pera Sep 1 '10 at 5:38