Introduction to Non-Archimedean Geometry

Lecture course, Fall 2020


Robert Delaunay 'Rythme no 1'
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.

What's new

Schedule

Please consult the syllabus [PDF].

Lecture notes change log, current version: [PDF]

Problem sets: 1 (due Oct 23), 2 (due Oct 30)

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 ---
Nov 2 ---
Nov 5 ---
Nov 9 ---
Nov 12 ---
Nov 16 ---
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Nov 30 ---
Dec 3 ---
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Holiday
Jan 7 ---
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Jan 28 ---

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