---
tag: YAH6MZVG7K
kind: concept
name: Rigid analytic space
msc: [14G22, 12J25]
sources:
  - url: https://en.wikipedia.org/wiki/Rigid_analytic_space
same_as:
  wikidata: Q7333799
  wikipedia: Rigid analytic space
---

## Summary

A rigid analytic space is the analogue of a complex analytic space over a complete non-archimedean field such as the [[$p$-adic numbers|BT6T702C28]].
It is glued from affinoid spaces, the maximal spectra of quotients of algebras of convergent power series, along a Grothendieck topology of admissible opens.
John Tate introduced the theory to describe elliptic curves with split multiplicative reduction as quotients of the multiplicative group.
It underlies $p$-adic uniformization and $p$-adic Hodge theory.

## Explanations

### L1 human

Over the $p$-adic numbers, the usual topology is totally disconnected: every disc splits into smaller disjoint open discs.
With this topology, functions defined piece by piece behave badly, and there is no good notion of analytic continuation.
Rigid analytic geometry fixes this by allowing only certain coverings, the admissible ones.
The closed unit disc, for instance, is not covered admissibly by the open unit disc and the circle $\abs{x} = 1$.
With this restriction, analytic functions on $p$-adic spaces behave much like holomorphic functions on complex spaces.