# 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:

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.