Like everyone else I am wildly cheering Peter Scholze’s new preprint constructing Galois representations attached to torsion classes — torsion classes! — in the cohomology of locally symmetric spaces for GL_n. I had been aspiring, and still do aspire, to develop enough of a global picture of how this works to write about it on the blog. But I’m happy to report that it looks like Persiflage, who’s somewhat closer to the subject than I am, is going to do it at his place. In his words:

This is mathematics which will, no question, have more impact in number theory than any recent paper I can think of. The basic intent of this post is to commit to future posts in which I will discuss the details.

At the risk of talking about stuff I dont understand yet, I’ll make one comment. It seems that a key technical development is Scholze’s ability to use the language of perfectoid spaces to talk about things like modular curves and modular varieties “at infinite level.” See how I reflexively put scare quotes there? It’s because, when I learned this stuff, it was customary to *pretend* to talk about infinite level, but really this was used as more of a metaphor; every actual argument I knew how to make took place in the pedestrian context of schemes of finite type over local and global fields. (Others may have been more daring, I don’t know.) Anyway, Scholze’s techniques seem to allow him to work fearlessly at the top of the tower, no scare quotes necessary, at which point new phenomena appear, phenomena which have implications even back at finite level.

(I am eager for this preliminary stuff to be corrected, refined, rebuked, and improved on in comments….!)

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I’m very ignorant on these topics, but you sound so enthusiastic about this that I feel obliged to ask: why do you say “torsion classes!”? I don’t know what a torsion class is (obviously lol) but what sounds most intriguing is your excitement at them having Galois representations attached to them… are those structures that don’t usually talk to each other? And why is it exciting that they do?

One possible way to view the pretend-versus-real distinction you draw might be that it (quite naturally) took some time for the philosophical ideas underlying Fontaine’s various period rings (e.g. it’s good to make Frobenius invertible even at the expense of making your rings nonnoetherian, deeply ramified extensions simplify things in many ways, etc.) to be digested and to find their natural place (perfectoid spaces) in more global settings.

Dear JSE, I agree with your remarks!