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To my knowledge, the strongest modularity lifting theorem known to date can be found in the preprint Potential automorphy and change of weight by Thomas Barnet-Lamb, David Geraghty, Toby Gee and Richard Taylor (accessible through their webpages).

A key idea in this work seems to be the introduction of a new local condition to impose on the liftings one wants to consider: the so-called potential diagonalizability. Since its original introduction by Richard Taylor, the notion of potential automorphy has similarly proven to be extremely useful, but the introduction of the latter notion seems to me much more easy to motivate than the former.

My question is: what motivated the introduction of the notion of potential diagonalizability?

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Graff's answer is (naturally) accurate. Historically I think it went something like this:

  • observation that the fact that "automorphy of a point on a component of a universal deformation ring implies automorphy of the whole component" (i.e. the general TWK method) could profitably be combined with the Harris tensor product trick to attack Sato--Tate, using the known structure of the FL and pot-BT def rings (this is the first BLGG paper).

  • independent observation that FL components contain "simple" points (the ones that imply potential diagonalizability for FL reps).

  • the observation that these two arguments could be combined with potential automorphy for ordinary representations (and thus crucially that one could move from ordinary to non-ordinary reps - this wasn't done in the original BLGG paper). Initially this was a rather complicated argument that was specific to the FL case (indeed Taylor's Ihara avoidance was used in the FL case rather than the ordinary setting). It was then realised that it was easier to do all level raising/lowering away from l in the ordinary case, and at this point it became clear that the argument in fact worked for potentially diagonalizable representations.

So I guess the answer is: there was an argument that was designed to work in the FL case, but once it was put into its final form, it was obvious that the correct condition was potential diagonalizability.

As far as I know, outside of the ordinary, FL and pot-BT (GL_2) cases, there aren't any other general cases in which one can prove potential diagonalizability at present. Of course there are some other concrete cases where one can, which can be useful e.g. for the weight part of Serre's conjecture (see the most recent BLGG paper).

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First, modularity theorems are not linearly ranked, and there are many modularity lifting theorems (of Kisin, Emerton, etc.) for $\mathrm{GL}(2)$ which can not be deduced from BLGGT.

Second, "potential diagonalizability" is the broadest class of representations for which the techniques of BLGGT (and the related paper of Barnet-Lamb, Gee, and Geraghty*) can be applied. This is fairly transparent if you actually read the papers. From this optic, potential diagonalizability should be thought of a technically convenient hypothesis which can be established in some key cases (Fontaine-Laffaille and ordinary representations, by BLGGT and the Geraghty's thesis respectively). Finally, the definition has the key property that it is invariant under base change (which is not true of being Fontaine-Laffaille, for example). For these reasons, it is hard to tell whether all potentially crystalline representations are potentially diagonalizable or not - I don't think anyone has any ideas about how to answer this question.

  • [Yes, I too was surprised, Geraghty does come alphabetically after Gee]
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  • $\begingroup$ Thanks for your comments, Graff. When you say "I don't think anyone has any ideas about how to answer this question" you mean my question or that of telling whether or not all potentially crystalline representations are potentially diagonalizable? $\endgroup$ – monodromy Nov 3 '11 at 14:02
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    $\begingroup$ I guess the latter; presumably the authors must have some idea of where the notion comes from... $\endgroup$ – monodromy Nov 3 '11 at 14:04
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    $\begingroup$ Dear Monodromy, The authors' motivations were (as far as I know) exactly as described by Graff Vynda-K: potential diagonalizability is a flexible notion, which incorporates already important notions such as Fontaine--Laffaille and ordinary, and which allows the Taylor--Wiles--Kisin method (as developed in BLGG and BLGGT) to work. But, as Graff Vynda-K also alludes to, the general answer to the question of which reps. are potentially diagonalizable is not yet known, and is among one of the major open questions in the field. Regards, Matthew $\endgroup$ – Emerton Nov 3 '11 at 17:11
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The question has already been well answered by Graff Vynda-K and TG but here are some extra comments. As mentioned in the intro to the paper, the main innovation is a new $n$-dimensional automorphy lifting theorem. In an automorphy lifting theorem, one starts with two congruent $l$-adic Galois representations $r$ and $r'$ with $r'$ being automorphic and one would like to deduce that $r$ is automorphic. In previous $n$-dim automorphy lifting theorems, one had to assume that locally at each prime above $l$, $r$ and $r'$ are either both ordinary or both Fontaine-Laffaille with the same Hodge-Tate weights. The improved lifting theorem in BLGGT allows one to assume the weaker condition that at each place above $l$, $r$ is either ordinary or Fontaine-Laffaille and similarly $r'$ is either ordinary or Fontaine-Laffaille. (In particular, one may be ordinary and the other FL, or both can be FL but with different Hodge-Tate weights.)

The proof of this improved result didn't make full use of the ordinary or Fontaine-Laffaille conditions. All that was needed was a certain ``diagonalizability'' property implied by the ordinary or FL conditions. It therefore made sense to give this property a name -- potential diagonalizability -- and to work with the class of representations that have this property.

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