# Introduction to Non-Archimedean Geometry

## Lecture course, Fall 2020

Robert Delaunay Rythme no 1 (1938)

Time: Mondays and Thursdays, 16:15-17:15

Venue: Zoom (Meeting ID: 861 9738 7959, e-mail me for the password)

Office hours: Tuesdays, 12-13:30 on Zoom (Meeting ID: 895 6034 0742)

#### Description

Non-Archimedean or rigid-analytic geometry is an analog of complex analytic geometry over non-Archimedean fields, such as the field of $p$-adic numbers $\mathbf{Q}_p$ or the field of formal Laurent series $k(\!(t)\!)$. It was introduced and formalized by Tate in the 1960s, whose goal was to understand elliptic curves over a $p$-adic field by means of a uniformization similar to the familiar description of an elliptic curve over $\mathbf{C}$ as quotient of the complex plane by a lattice. It has since gained status of a foundational tool in algebraic and arithmetic geometry, and several other approaches have been found by Raynaud, Berkovich, and Huber. In recent years, it has become even more prominent with the work of Scholze and Kedlaya in $p$-adic Hodge theory, as well as the non-Archimedean approach to mirror symmetry proposed by Kontsevich. That said, we still do not know much about rigid-analytic varieties, and many foundational questions remain unanswered.

The goal of this lecture course is to introduce the basic notions of rigid-analytic geometry. We will assume familiarity with schemes.

#### Schedule

Please consult the syllabus [PDF] (outline slightly outdated).

Lecture notes: change log, current version: [PDF]

Problem sets: 1 (due Oct 23), 2 (due Nov 1), 3 (due Nov 8), 4 (due Nov 18), 5 (due Nov 29), 6 (due Dec 6), 7 (due Dec 13), 8 (due Dec 20), 9 (due Jan 10), 10 (part I) and (part II) (due Jan 31)

All homework problems in one file: [PDF]

Selected solutions: Problem 3 on PS 8

Term paper guidelines and possible topics: [PDF]

Review of blowups (office hours on Nov 10): [PDF]

 Oct 15 Motivation and overview Oct 19 Valuations and non-Archimedean fields (I) Oct 22 Valuations and non-Archimedean fields (II) Oct 26 The Tate algebra (I) Oct 29 The Tate algebra (II) Nov 2 Affinoid algebras and spaces Nov 5 Sheaves, sites, and topoi Nov 9 G-topologies Nov 12 No lecture Nov 16 Affinoid subdomains Nov 19 The admissible topology Nov 23 Tate acyclicity Nov 26 Rigid spaces Nov 30 Analytification Dec 3 Properties of morphisms Dec 7 The Tate curve (I) Dec 10 The Tate curve (II) Dec 14 Formal schemes (I) Dec 17 Formal schemes (II) Dec 21 Rigid-analytic generic fiber Holiday Jan 7 Formal models Jan 11 Admissible blowups Jan 14 Raynaud's equivalence (I) Jan 18 Raynaud's equivalence (II) Jan 21 Rigid-analytic spaces revisited Jan 25 Riemann-Zariski spaces Jan 28 Riemann-Zariski spaces and valuations

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