Two questions and a story

\bullet Let f be some cuspidal Hecke eigenform, with associated Galois representation \rho_{f}:G_{\mathbf{Q}}\to \mathrm{GL}_2(\overline{\mathbf{Q}_p}). A notorious conjecture of Greenberg asserts that if \rho_{f}|G_{\mathbf{Q}_p} is abelian (i.e. is a direct sum of characters), then f is a CM form, or equivalently \rho_f is induced from a character. At some point I was talking about this with Barry Mazur, and he suggested a possible generalization:

Let \rho:G_{\mathbf{Q}} \to \mathrm{GL}_n(\overline{\mathbf{Q}_p}) be an irreducible geometric Galois representation. Suppose that \rho|G_{\mathbf{Q}_p} is a direct sum of characters. Is \rho induced from a character?

Emerton found a nice argument which proves Greenberg’s conjecture conditionally on some kind of p-adic variational Hodge conjecture. Is there any similar evidence for this question in some higher-dimensional cases?

\bullet Do separated etale maps of schemes satisfy effective descent with respect to fpqc covers? This is known if one restricts to quasi-compact separated etale maps. An analogous result is true for perfectoid spaces.

\bullet Here’s a funny story I heard from Glenn Stevens a while back:

At some point in the early ’90s, before he announced his proof of Fermat, Wiles came to Boston and gave a seminar talk at BU. He spoke about what is now known as the Greenberg-Wiles duality formula. However, he didn’t mention his main motivations for this formula. The upshot is that Stevens came away from the talk with the sad feeling that Wiles had lost his touch.

Report from Tucson

Just back from the 2017 Arizona Winter School on perfectoid spaces.  First of all, I should say that everything was impressively well-organized, and that the lecturers did a fantastic job, especially considering the technical weight of this material. (Watch the videos if you don’t believe me.)  Jared Weinstein, in particular, has an almost supernatural ability to make a lecture on some technical thing feel comforting.

Now to the jokes.

  • In his opening lecture, Scholze called perfectoid spaces a “failed theory”, on account of his inability to completely settle weight-monodromy. “You see, I’m Prussian, and when a Prussian says he wants to do something, he really feels responsible for doing it.”
  • Audience member: “Why are they called diamonds?”
    Scholze: “[oral explanation of the picture on p. 63 of the Berkeley notes]”
    Weinstein: “Also, diamonds are hard.”
  • Anon.: “When you’re organizing a conference, the important thing is not to give in and be the first one who actually does stuff.  Because then you’ll end up doing everything!  Don’t do that!  Don’t be the dumb one!”
    Me: “Didn’t you organize [redacted] a couple of years ago?”
    Anon.: “Yeah… It turned out that Guido Kings was the dumb one.”
  • Mazur: “It just feels like the foundations of this area aren’t yet… hmm…”
    Me: “Definitive?”
    Mazur: “Yes, exactly.  I mean, if Grothendieck were here, he would be screaming.”
  • “Do you ever need more than two legs?”
  • During the hike, someone sat on a cactus.
  • Finally, here is a late night cartoon of what a universal cohomology theory over \mathbb{Z} might look like (no prizes for guessing who drew this):