Perfectoid reading group (Winter 2014)


We meet in Eckhart 207 on Tuesdays from 3:00pm-5:00pm (see below for the exact schedule), with a break for tea. If you want to be added to the mailing list, e-mail me (Sean).

Outline
The focus will be on understanding parts of Scholze's "On torsion in the cohomology of locally symmetric varieties" in the case of modular curves. Roughly, we hope to spend about half the time on the Hodge-Tate period map (Chapter III), and after that about half the time on p-adic modular forms (Chapter IV), all within the context of modular curves. Note that this is the student number theory seminar for this quarter.

Schedule
Meetings take place Tuesdays 3:00-5:00pm in Eckhart 207, with a break for tea.

7 Jan. -- Organizational meeting

14 Jan. -- Sean, Tate's "p-Divisible Groups" part I. notes

21 Jan. -- Sean, Tate's "p-Divisible Groups" part II. notes

28 Jan. -- Preston, The Lubin-Tate perfectoid space. notes

4 Feb. -- Sean, Drinfeld's proof of Serre-Tate. notes

11 Feb. -- Fan, the Drinfeld upper halfplane. notes

18 Feb. -- NO MEETING; Break for MSRI Hot Topics: Perfectoid Spaces workshop (if you're not going then you can still watch lectures online!)

25 Feb. -- Dan, the Grothendieck-Messing period map and moduli of p-divisible groups. notes

4 Mar. -- Matt, the Hodge-Tate period map. notes

11 Mar. -- Dan,

Resources
Scholze -- On torsion in the cohomology of locally symmetric varieties.

Scholze IHES lecture.

Scholze's 2013 MSRI lectures, videos available at the meeting website. Most relevant for us is the final lecture of the series, similar in contents to the IHES lecture above. The others provide useful background on zeta functions of Shimura varieties, with a focus on modular curves.

Outline of Scholze's seminar at Bonn.

Frank Calegari's blog posts on the subject: Parts 0, 1, 2, 3, 4. Note these are a bit tangential to our goals -- Calegari focuses mostly on Chapter V which reduces the problem of finding Galois representations associated to torsion classes to the problem of interpolating Hecke eigenvalues in the cohomology of Shimura varieties by Hecke eigvenvalues of cusp forms, whereas our focus will be on this interpolation problem and understanding some of the perfectoid geometry of Shimura varieties that goes into it (in the case of modular curves).

Jared Weinstein's 2013 Arizona Winter School lectures on "Modular curves at infinite level," videos and notes available at the meeting website.

Scholze's 2012 CDM survey on perfectoid spaces.

Scholze, Weinstein -- Moduli of p-divisible groups.

Weinstein -- Semistable models for modular curves of arbitrary level.

Scholze -- p-adic Hodge theory for rigid-analytic varieties.

Weinstein IHES lecture on Moduli of p-divisible groups.

English translation of Boutot-Carayol on p-adic uniformization of Shimura curves (cf. also Boutot's Bourbaki talk and Kevin Buzzard's notes).

Frank Calegari's AWS lecture notes on p-adic modular forms. In particular, the section on canonical subgroups provides some good examples.

Tate -- p-Divisible Groups

Koshikawa's Keywords
Koshikawa put together the following informal list of keywords to help us stay pointed in the right direction (most references below are listed with links in the resources above):

Modular curves -- adelic view, compactification, usual integral model and Neron models, ordinary and supersingular points

Ordinary locus -- Hasse invariant and affineness, Frobenius lift on the formal completion, level structures
Remark: It's possible to construct a correct perfectoid space for ordinary locus at this stage.

Completion along a supersingular point -- Formal group laws and p-divisible groups, Serre-Tate theorem, Lubin-Tate space and their coverings, Lubin-Tate space at infinite level (cf. Weinstein references)
Remark: Logically we don't need to prove LT is perfectoid because this follows from that modular curve is perfectoid (the other direction does not work)

p-adic Hodge theory -- classical Hodge-Tate decomposition, (almost integral) Hodge-Tate filtration (cf. Scholze's CDM survey paper), classification of p-divisible groups (cf. Scholze-Weinstein)

Faltings isomorphism -- Drinfeld space and its coverings, Faltings isomorphism at infinite level (cf. Scholze-Weinstein)

How to "glue" ordinary locus and supersingular locus as perfectoid spaces (Chapter III) -- canonical subgroups (deformations of p-divisible groups and almost mathematics), anticanonical tower (and Tate's normalized trace), changing level structures (tilting and almost purity), extension to the whole space with Hodge-Tate period map.
Remark: modular curve is somewhat easier than Siegel case.

p-adic modular forms (Chapter IV) -- "classical" theory (Hasse invariant trick), completed cohomology, comparison theorems (Faltings, Scholze), fake Hasse invariant and conclusion.