New Results In Modularity, Christmas Update

It’s a Christmas miracle! Keen watchers of this blog will be happy to learn that the 10 author paper discussed here and here is now available. (And just in case you also missed it, you can also find the other modularity result discussed here at this link.)

You may ask — what took so long? Well, it’s quite long, as you might notice. Also, as it turns out, one version of this paper was almost ready quite some time ago, but then quite a bit of time was spent improving the ordinary local-global compatibility statements and thus the ordinary modularity lifting theorems so that they would apply to small p. The hypothesis in the Fontaine–Laffaille chapter is that p > n^2, which is not ideal if one wants to take n = 2 and p = 2 or 3 (which indeed one might have occasion to do), but now the hypothesis in the ordinary section is much less restrictive, so yay for that.

The paper relies on the (as yet unavailable) work of Caraiani-Scholze; I hope to be able to have a post explaining a little bit more about that paper relatively soon.

Finally, I wanted to make one remark about credit. I think with an almost 200 page paper (*) that proves a number of quite nice results, there’s enough credit to go around among 10 different authors. On the other hand, I think it’s worth pointing out (perhaps even incumbent upon me to say) that the younger authors of this paper did more than their fair share of work — both creatively, intellectually, and practically.

I hope you enjoy reading this paper as you tuck into your plum pudding, unless you happen to be one of those peculiar people who doesn’t much care for the thought of a pudding consisting of dried fruits held together by eggs, breadcrumbs, cognac, and a bunch of fat from beef kidneys, in which case your soul must be even more desperately in need of nourishment and thus you will be even more in want of a paper such as this.

(*) to be fair, almost 200 pages using a LaTeX style file with generously wide margins — it’s probably only more like 150-160 in the standard format.

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7 Responses to New Results In Modularity, Christmas Update

  1. ME says:

    Congratulations! A Christmas miracle indeed! (I just saw this post a short while ago; if I’d known earlier, I would have congratulated you in person!)

  2. JC says:


  3. Nice to see this arrive. But… what sort of reference is this??

    “David Hansen, Minimal modularity lifting for GL2 over an arbitrary
    number field, To appear in Math. Res. Lett., September 2012.”

    • I think I can guess what happened — originally, someone had a reference in their .bib file to the arXiv version of this paper (probably with the bibtex entry that one can create automatically from the arXiv, which includes the month). Then, the paper was accepted, and a note was added to that effect. This brings up a general BibTeX question. Suppose you reference a pre-print which is “to appear.” Presumably the correct format is @unpublished. What is one supposed to put in the “YEAR” entry? The year the paper was written, the year it was accepted, or the year that you guess it is going to appear? It seems as though the first option makes the most sense, but then you end up with references that look like the one you are complaining about. Or maybe the answer is that one just deletes the year entirely?

      • The lack of arXiv references for preprints is also slightly annoying. For the purposes of bibliographic metadata and citation tracking etc (as well as user-friendliness) it is very helpful to include them (and even more so when the paper is published, for those who don’t have access to the journal). If the entry I mentioned had an arXiv number, it would be more clear why the year is 2012, and that it isn’t a typo, or some “to appear” paper that never actually appeared. Finite group theory in the 80s has such papers, and sometimes such references propagated well past when it should have been clear the paper wasn’t turning up.

  4. Pingback: Local-global compatibility for imaginary quadratic fields | Persiflage

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