18.727 Notes

The primary reference is Rigid analytic geometry and its applications, by Fresnel and van der Put. It will be cited throughout this page and these notes as [FvdP]. The secondary reference is Non-archimedean analysis, by Bosch, Güntzer, and Remmert. It will be cited throughout this page and these notes as [BGR].

The green ball indicates the current position of the course. I'll be trying to keep a little bit ahead of the green ball, but so far I haven't been managing so well.

I'll also collect here some other useful links, such as:

Online version of Peter Schneider's book Nonarchimedean Functional Analysis (self-explanatory)
My arXiv preprint Full faithfulness for overconvergent F-isocrystals (for the exercises on Quillen-Suslin for Tate algebras)

I am working on a list of optional topics for later in the course: here is the current version [tex, dvi, ps, pdf]

I plan to amend the notes posted below to incorporate light corrections; more serious corrections found during the semester are being posted as errata in later notes. I'm hoping later on to redact everything into a coherent and relatively bug-free narrative; if energy permits, that might ultimately coalesce into a book, but don't hold your breath. Anyway, corrections and comments are always welcome.

Introduction

Example: the Tate curve (amended 16 Sep) [tex, dvi, pdf, ps]

A little p-adic functional analysis

Part 1 of 2 (amended 16 Sep) [tex, dvi, pdf, ps]
Part 2 of 2 (amended 16 Sep) [tex, dvi, pdf, ps]
See also: [FvdP, Chapter 1], and various sections in [BGR, Chapter 2].

Affinoid algebras

Tate algebras [tex, dvi, pdf, ps]
Affinoid algebras and their spectra [tex, dvi, pdf, ps]
More on affinoid algebras and their spectra (amended 4 Oct 04) [tex, dvi, pdf, ps]
Addendum [tex, dvi, pdf, ps] See also: [FvdP, Chapter 3] and [BGR, Sections 3.8, 5.1 and 6.1].

The projective line

Subsets of the projective line, part 1 [tex, dvi, pdf, ps]
Subsets of the projective line, part 2 (includes G-topologies) [tex, dvi, pdf, ps]
More on $G$-topologies, part 1 of 2 [tex, dvi, pdf, ps]
More on $G$-topologies, part 2 of 2 [tex, dvi, pdf, ps]
See also: [FvdP, Chapter 2].

Affinoid spaces and their geometry

The $G$-topologies of an affinoid space [tex, dvi, pdf, ps]
Coherent sheaves and Tate's acyclicity theorem (amended 25 Oct) [tex, dvi, pdf, ps]
See also: [FvdP, 4.1 and 4.2].

Rigid spaces and sheaf cohomology

Rigid analytic spaces [tex, dvi, pdf, ps]
More on coherent sheaves [tex, dvi, pdf, ps]
Kiehl's finiteness theorems [tex, dvi, pdf, ps] See also: [FvdP, Chapter 4].

The Berkovich perspective

Berkovich spaces for dummies [tex, dvi, pdf, ps]
More errata [tex, dvi, pdf, ps] (note: the first errata are in the first p-adic cohomology handout, and some of these are errata of errata!)
See also: [FvdP, Chapter 7] and Berkovich's papers (more precise references in the notes).

p-adic cohomology

p-adic cohomology, part 1 [tex, dvi, pdf, ps]
[tex, dvi, pdf, ps]
See also: [FvdP, 7.5-7.7] and references in the notes.

Formal groups

The Lubin-Tate moduli space [tex, dvi, pdf, ps]

Periods on the Lubin-Tate moduli space [tex, dvi, pdf, ps]
See also: references in the notes, and the Gross-Hopkins article from Bulletin of the AMS. (That's the short summary; the longer detailed article is not available online.)

p-adic uniformization

p-adic uniformization and Shimura curves [tex, dvi, pdf, ps]
See also: [FvdP, Chapter 5] and references in the notes.