Introduction to Non-Archimedean Geometry 1000-1M20WGN

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 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 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.

Tentative outline:

I. Motivation and overview; topology of p-adic numbers; valuation rings
II. Topological & adic rings
III. Formal schemes
IV. Tate algebras
V. G-ringed spaces & the admissible topology
VI. Rigid-analytic spaces
VII. Examples of rigid-analytic spaces. The Tate curve
VIII. Raynaud's approach
IX. Applications

Additional topics:
a. Huber's theory of adic spaces
b. Berkovich spaces
c. Riemann--Zariski spaces
d. Nagata's compactification theorem