Wear Project

The working group will be led by Gabriel Dorfsman-Hopkins, Ravi Fernando, and Jackson Morrow. To participate in this working group, send an email to one of these organizers.

Both students and faculty are encouraged to participate in the working groups.

The project meets at 1:30-5 pm at Evans 947.

Lecture Series:

Lecture 1: Perfectoid spaces and weight-monodromy

Scholze's original motivation for developing the theory of perfectoid spaces was to prove Deligne's weight-monodromy conjecture. In this largely expository lecture, we motivate and state the conjecture, then give a sketch of Scholze's approach in the case of a hypersurface in projective space. We then point out the nice features that made this case work and explain what needs to be done to extend his approach further.

Lecture 2: Perfectoid covers of abelian varieties

We explain how to prove weight-monodromy for complete intersections in abelian varieties. This builds off of joint work with Blakestad, Gvirtz, Heuer, Shchedrina, Shimizu, and Yao on constructing perfectoid covers of abelian varieties, and joint work with Heuer on tilting these perfectoid covers. We then point out what nice features made this generalization possible, and pose some questions about potential further generalizations.

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Project Questions:

Question 1: Perfectoid covers of.......?

Here's the basic dream that makes this all work: given a variety X over a local field k, can we find a perfectoid field K extending k, and a perfectoid space X--->X_K whose tilt lies over a variety X', and such that the construction is "natural enough" to pass cohomological information from X to X'? This is what Scholze does for (complete intersections in) toric varieties, and the main goal of the second talk is to explain this for abelian varieties.

Here's a possible approach: A scheme over a field which is perfect and of characteristic p comes equipped with a relative Frobenius morphism. One doesn't typically expect to be able to lift this (though it is possible for toric varieties and for ordinary abelian varieties, which is useful in the above!). That being said, for any abelian variety, one can lift multiplication by p, which factors over relative Frobenius! This is essentially why we get a perfectoid cover. Inspired by this example we ask: when can we find a cover of the the relative Frobenius which lifts, and how nicely/naturally can this be done?

One question which I would like to understand is what conditions must be placed on the map and/or the lift to ensure that we can pass around cohomological information. In my thesis, the lifts were either finite ètale or a composition of finite ètale and radical. This seems too restrictive though, how much can it be weakened?

One explicit place to start thinking about the overall question is the case of K3 surfaces. There doesn't seem to be much hope if we restrict the cover to also be a K3 surface (can someone make this precise?), but that doesn't mean we should give up. I'd expect it to be actually possible to say something meaningful about Kummer surfaces, by comparing them to the abelian surface which covers them. Tilting these abelian surfaces should also tilt the Kummer surfaces, which would tell us how to pair them up. And once something concrete is said there, maybe it will shed some light on how to proceed more generally.

Question 2: Perfectoid quotients of universal covers of abelian varieties.

(From Piotr Achinger via Ben Heuer)

Given an abelian variety over a characteristic 0 perfectoid field, we take the inverse limit over all finite etale covers (or the cofinal system consisting of multiplication the multiplication maps [n]:A-->A for all n) to get a pefectoid space A ---> A, which is a Galos with Galois group isomorphic to the fundamental group of A . For which subgroups H of π_1(A) is A/H perfectoid?

Thinking about this first in the cases of ordinary good reduction and/or Tate curves should be interesting, and a good way to get more comfortable with the general ideas behind building perfectoid covers of abelian varieties.

Question 3: Universal covers of curves.

(A hard question from Ben Heuer)

(see Section 6 of https://arxiv.org/pdf/2105.12604.pdf for more details)

Given a curve, we can form a perfectoid cover by embedding it in its Jacobian and pulling back along the perfectoid cover from Section 2 (see section 5.3 of https://arxiv.org/abs/1804.04455). Ben showed that if two abelian varieties are "p-adically close", then their perfectoid covers are isomorphic. Is the same true for curves? Note that this is already both interesting and unclear in the good reduction case! (Surprised? I expected that if two curves have the same special fiber, we can generate the whole special fiber of the tower just by looking at the entire picture down there. But pulling back along will introduce singularities on the special fiber of the curve.)

But anyway, we could either try to think about the good reduction case, or perhaps could try thinking about Mumford curves. Could also try to look only at the pro-etale cover of a Mumford curve by looking only at other covers by Mumford curves - that would likely be easier and could still be interesting?

Note that this question is likely quite hard. Also note that this isn't the only way to form perfectoid covers of curves, and I don't claim it's the best way either. Maybe we could try to think of other options?

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We could also spend time working through some of the background on weight-monodromy. Understanding where it's coming from, how the weight and monodromy filtrations are constructed, and understanding Scholze's approach are all interesting and worth doing. I can suggest references for those if there's interest.

Reading list:

Scholze's thesis https://arxiv.org/abs/1111.4914 with an emphasis on sections 8 and 9. Also his opening lecture at the 2017 winter school was nice: https://www.math.arizona.edu/~swc/aws/2017/

My thesis https://escholarship.org/uc/item/1ww154gc is where most of the stuff I'll talk about is written up. I'm working on a briefer version, but it probably won't be ready by the workshop. This is the place to read about question 1

Perfectoid covers of abelian varieties https://arxiv.org/abs/1804.04455 gives a succinct construction of the perfectoid spaces that will be featured, but we'll need a different construction in the talk. (We need formal models!) Section 5.3 on perfectoid covers of curves will be useful for question 2.

Ben Heuer's paper on pro-etale uniformization of abelian varieties https://arxiv.org/abs/2105.12604 has really nice results which I think clarify how one should think about these covers (specifically if abelian varieties are "p-adically close" their perfectoid covers will be isomorphic. Corollaries 1.5 and 1.6 are specifically very nice. This will be useful for question 3.