# Fall 2014 MATH 274 001 LEC

Topics in Algebra
Schedule:
SectionDays/TimeLocationInstructorCCN
001 LECTuTh 11-1230P 740 EVANSSCHOLZE, P54488
Units/CreditFinal Exam GroupEnrollment
4NONELimit:39 Enrolled:4 Waitlist:0 Avail Seats:35 [on 10/09/14]
Discussions:
SectionDays/TimeLocationInstructorCCN
101 VOLTu 930-1030A 740 EVANSWEINSTEIN, J S

Subject: $p$-adic geometry

Instructor: Peter Scholze

Office: 1063 Evans Hall

Office Hours:

Scheduling Notes: Note that the class will be held in 740 Evans (not 285 Cory, as shown earlier), that the first class meeting will be September 2, and that there will be auxiliary weekly lectures by Jared Weinstein, as shown above.

Prerequisites: We will assume some background in algebraic geometry and Dieudonné theory. Tutorials held by Jared Weinstein will recall some of the intervening notions.  (See Scheduling Notes above, and Background Lectures, after the description of the main course below.)

Description: Originally defined by Drinfel'd, and then used extensively by L. Lafforgue and V. Lafforgue, moduli spaces of shtukas have proved to be a powerful tool in the study of the Langlands correspondence over function fields. It is an important question whether something similar could work over number fields.

In this course, we want to sketch a strategy to define moduli spaces of local shtukas over mixed-characteristic fields such as $\mathbb{Q}_p$ (leaving open the problem of assembling these spaces for varying primes $p$). These spaces should generalize Rapoport-Zink spaces, as well as the conjectural theory of local Shimura varieties that has recently been suggested by Rapoport-Viehmann. Notably, the spaces of local shtukas should overcome the 'minuscule' condition inherent in the theory of Shimura varieties, so that they can be seen as very general $p$-adic period domains (which exist even in situations where Griffiths transversality is a nontrivial condition). We wll start by reviewing the theory of (local and global) shtukas over function fields. Next, we will define shtukas in mixed characteristic in an absolute setup; this is closely related to the notion of Breuil-Kisin modules, and has been the subject of recent investigations of Fargues, some of whose results we will recall.

To set things into perspective, we will spend some time detailing the case of $p$-divisible groups, explaining the equivalence between $p$-divisible groups and certain shtukas (due to Breuil, Kisin, and Fargues). We will then use the joint work with Weinstein to identify Rapoport-Zink spaces with moduli spaces of shtukas.

We will then be able to define general spaces of shtukas. They will turn out to be somewhat esoteric objects, called 'diamonds', so quite a bit of time will be spent on explaining the definition of diamonds. In particular, we will explain how they overcome the problem that a general space of shtukas should live over 'a product of several copies of $\mathrm{Spa}\ \mathbb{Q}_p$' (taken over some absolute base '$\mathbb{F}_1$'), by giving a highly nontrivial definition of 'a product of several copies of $\mathrm{Spa}\ \mathbb{Q}_p$' in a way that makes Drinfel'd's lemma true.

If time permits, we will try to (conjecturally) understand the étale cohomology of these spaces, and their relation to the local Langlands correspondence.

Background lectures by Jared Weinstein: These weekly meetings are for graduate students who wish to follow Peter Scholze's Math 274 course on p-adic geometry.  We will discuss  Drinfeld's "elliptic modules" -- this is a formalism which encompasses cyclotomic fields, elliptic curves, and Drinfeld modules, which appear in characteristic p.  In the last case, Drinfeld modules (and the related concept of "shtukas") led to a really striking achievement -- L. Lafforgues' proof of Langlands' conjectures for a function field. Algebraic geometry background will be provided as necessary.

Required Text: