Just returned from a workshop on “Arithmetic of Shimura varieties” at Oberwolfach. Some scattered recollections:
- Gabber wasn’t there, but there were some Gabberesque moments anyway. In particular, during Xuhua He’s talk, Goertz observed that a point is an example of a Deligne-Lusztig variety, so any variety is a union of Deligne-Lusztig varieties. Gotta be careful…
- The food was about the same as usual. Worst Prize was tied between two dishes: a depressing vegetable soup which somehow managed to be flavorless and bitter simultaneously, and a dessert which looked like a lovely innocent custard but tasted like balsamic vinegar. The best dishes were all traditional German fare.
- Best Talks (in no particular order): Jean-Stefan Koskivirta, Miaofen Chen, Ben Howard, Timo Richarz.
- Apparently this paper can be boiled down to a page or two.
- There was (not surprisingly) some late-night discussion of the Stanford Mystery. [Redacted] proposed a theory so mind-bogglingly outrageous that it certainly won’t fit in this margin.
- “Fun was never really my goal.” – A representative UChicago alum.
- On Thursday it snowed, and a snowball fight broke out after dinner. This was a lot of fun, but I’m still glad we didn’t follow Pilloni’s suggestion of a match between Team Europe (Pilloni, Stroh, Morel, Anschutz, Richarz, Mihatsch, etc.) and Team USA (me).
- Here’s an innocent problem which turns out to be pretty tricky. Let be a (separated, smooth) rigid analytic space over , and let be a map from a perfectoid space which is a -torsor for some profinite group . In shorthand, you should think that with G acting freely (this is all literally true in the category of diamonds). It’s easy to cook up examples of this scenario: for instance, you can take and , so then is a torsor for the group . However, there are also much more complicated examples which arise in nature. In particular, if is a Rapoport-Zink space or abelian-type Shimura variety at some finite level, and is the associated infinite level perfectoid guy over it, then we’re in the situation above, with open in the -points of some auxiliary reductive group.
Anyway, supposing we’re in the situation above, we can ask the following complementary questions:
Q1. Suppose that is affinoid perfectoid. Does this imply that is an affinoid rigid space?
Q2. Suppose that is an affinoid rigid space. Does this imply that is affinoid perfectoid?
It seems like both of these questions are actually really hard! For Q1, we can (by assumption) write for some perfectoid Tate-Huber pair , and then one might guess that coincides with . There is certainly a map , but now one is faced with the problem of showing that is “big enough” for this map to be an isomorphism. This can be reduced to any one of a handful of auxiliary problems, but they all seem hard (at least to me). For instance, as a warmup one could try to prove either of the following implications:
W1. Under the hypothesis of Q1, vanishes.
W2. Under the hypothesis of Q1, is killed by a fixed power of
Both of these conclusions would certainly hold if we already knew that was affinoid: the first is just (a consequence of) Tate acyclicity, while the fact that is killed by some power of for smooth affinoids is a non-trivial theorem of Bartenwerfer. But I have totally failed to prove either W1 or W2.
In any case, the essential point with Q1 seems to be the following. If is some open subgroup, then will always have plenty of elements, and indeed taking the direct limit as shrinks recovers . But the obstruction to lifting an element of to an element of is a torsion class in , and the latter group seems hard to control.
For Q2, there is maybe a slightly clearer path through the forest: it would follow from the following conjecture, which I explained during my talk in the workshop.
To set things up, let be any uniform Tate-Huber pair over , and let be the associated pre-adic space. Let denote the site given by perfectoid spaces over with covers given by v-covers, and let and be the obvious structure sheaves on . Set and , so the association is an endofunctor on the category of uniform Tate-Huber pairs over . One can check that breve’ing twice is the same as breve’ing once, and that the natural map induces an isomorphism of diamonds. If is a smooth (or just seminormal) affinoid -algebra for some , or if is perfectoid, then breve’ing doesn’t change .
Conjecture. Let be a uniform Tate-Huber pair over such that every completed residue field of is a perfectoid field. Then is a perfectoid Tate ring.
Aside from disposing of Q2, this conjecture would settle another notorious problem: it would imply that if is a uniform sheafy Huber ring and is a perfectoid space, then is actually perfectoid.
It may be instructive to see an example of a non-perfectoid (uniform) Tate ring which satisfies the hypothesis of this conjecture. To make an example (with ), set , and let with the obvious topology. Set , so there are natural maps . Then and are perfectoid, but isn’t: the requisite -power roots of mod don’t exist. Nevertheless, every completed residue field of is perfectoid (exercise!), and the map induces an isomorphism .
OK, this bullet point turned out pretty long, but these things have been in my head for the last couple months and it feels good to let them out. Besides, Yoichi Mieda asked me about Q1 during the workshop, so despite the technical nature of these questions, I might not be the only one who cares.
- Oberwolfach continues to be one of the best places in the world to do mathematics.
Thanks to the organizers for putting together such an excellent week!