**1**

vote

**2**answers

142 views

### Is there a notion of pure dimension for Berkovich analytic space?

For affinoid spaces the definition is similar to algebraic geometry, what about general analytic spaces? I can't find a reference about it. If yes then is the analytification of a variety of pure ...

**5**

votes

**0**answers

110 views

### $p$-adic uniformisation of abelian varieties

In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement:
Let $A$ over $\mathbf{Q}_p$ be an abelian variety ...

**5**

votes

**1**answer

151 views

### Rigid analytic geometry in characterstic 0 vs positive characteristic

This question is motivated purely by curiosity. In algebraic geometry there is a major distinction between the world of characteristic $0$ and that of characteristic $p > 0$ with different methods, ...

**5**

votes

**1**answer

101 views

### Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)

In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid.
Could you ...

**13**

votes

**0**answers

258 views

### Vanishing of rigid cohomology for affine varieties

Let $k$ be a perfect field of positive characteristic and denote by $K$ the field of fractions of the ring of Witt vectors over $k$.
Question: If $X$ is an affine variety over $k$, do the rigid ...

**0**

votes

**2**answers

150 views

### Berthelot functor, rigid analytic space

If $X=\operatorname{Spec} A$, where $A$ is a noetherien, complete local ring, with a finite residual field $\mathbb{F}_p$. We can associate to $A$ a rigid analytic space with two different ways, we ...

**10**

votes

**0**answers

181 views

### Detecting $k$-affinoid spaces by vanishing cohomology

The property of being an affine scheme can be tested against all quasi-coherent sheaves in the following sense: a noetherian scheme $X$ is affine iff $H^i(X,\mathcal{F}) = 0$ for all quasi-coherent ...

**2**

votes

**2**answers

180 views

### Product of reduced affinoid spaces over a field is reduced (reference request)

Let $K$ be a field of characteristic zero complete with respect to a non-Archimedean absolute value. Suppose that $A$ and $B$ are two affinoid $K$-algebras. I'd like a reference that will answer the ...

**8**

votes

**1**answer

190 views

### why don't (can't?) we sheafify the structure presheaf of an adic space

In the definition of an adic space, usually there is a presheaf defined by first saying what it is on a particular basis of the topology of the underlying space, the so called rational subsets. One ...

**8**

votes

**3**answers

323 views

### Trivialisation of vector bundles on Stein spaces

Does every vector bundle on a Stein space have a finite local trivialisation?
Definitions:
Stein space means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of ...

**3**

votes

**1**answer

240 views

### Paper of Boutot-Carayol in `Courbes modulaires et courbes de Shimura'

I am trying to obtain a copy of the following
J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les
théorèmes de Čerednik et de Drinfel'd , Astérisque No. 196-197 ...

**4**

votes

**1**answer

133 views

### Topology of ring of global sections of finite union of affinoid opens in a rigid analytic space

Let $X$ be a rigid analytic space over a non-Archimedean field $k$. If $U_1,\ldots,U_n\subseteq X$ are affinoid opens, then it's usually not clear whether or not the admissible open ...

**14**

votes

**0**answers

845 views

### How is an $S^1$-equivariant elliptic cohomology theory affected as we continuously vary the underlying elliptic curve?

Grojnowski constructs a $S^1$-equivariant cohomology theory over a complex elliptic curve $E$, designed to trivially satisfy: $$E^*_{S^1}(pt) = E$$
The functor $E^*_{S^1}(-)$ takes in a space $X$ ...

**5**

votes

**1**answer

156 views

### Berkovich stalk versus rigid analytic stalk

Let $A$ be a strictly affinoid algebra. Let $X^{Ber}$ bet its Berkovich spectrum and $X^{Tate} = \operatorname{Sp} A$ its affinoid variety in the sense of rigid analytic geometry. Let $\mathfrak{m} ...

**0**

votes

**0**answers

131 views

### Fiber products of adic spaces

In the notes from Peter Scholze's class at Berkeley he makes the following remark: "Let us call a Huber pair $(A, A^+)$ admissible if $A$ is finitely generated over a ring of definition $A_0 \subset ...

**1**

vote

**0**answers

72 views

### Relative nonarchimedean disks and annuli

Let $A$ be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring $A_0$ which is adic and has finitely generated ideal of definition.
Is there a good notion of closed disk of ...

**2**

votes

**1**answer

278 views

### The rigid-analytic open disk

Let $K$ be a local field and $D_K$ the open unit disk, considered as a rigid space or adic space over $K$. What is the algebra of analytic functions on $D_K$? Proposition 1.1 of this article describes ...

**22**

votes

**1**answer

770 views

### function field analogy and global/absolute geometry

The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...

**4**

votes

**1**answer

333 views

### Iwasawa logarithm and analytic continuation

I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$.
...

**11**

votes

**2**answers

551 views

### Étale homotopy type of non-archimedean analytic spaces

The following is likely all obvious to the experts. But since the field looks tricky to an outsider, maybe I may be excused for asking anyway.
I am wondering about basic facts of what would naturally ...

**6**

votes

**0**answers

150 views

### Picard group of Drinfeld upper half space

Let $K$ be a $p$-adic field and $\Omega^{(n)}_K$ the $n$-dimensional Drinfeld upper half space over $K$ (which is a rigid analytic space over $K$).
Is the Picard group of $\Omega^{(n)}_K$ known? ...

**7**

votes

**1**answer

346 views

### What are the automorphisms of a perfectoid Tate algebra?

Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a_1T+a_2T^2+\cdots$, where $a_1\in ...

**5**

votes

**1**answer

199 views

### Are admissible open subsets of an affinoid space of countable type?

A rigid analytic space $Y$ over a complete non-archimedean valued field $k$ is said to be of countable type if it has a countable (possibly finite) admissible covering by affinoids over $k$.
...

**2**

votes

**1**answer

196 views

### Is X_0(p) a Mumford curve over $Q_{p^2}$

Let $p$ be a prime number and $X_0(p)/\mathbf{Q}$ be the classical modular curve for
$\Gamma_0(p)$. Let $\tilde{X}_0(p)/\mathbf{Z}$ be the projective arithmetic surface corresponding to the ...

**11**

votes

**0**answers

584 views

### A functor of points approach to Berkovich analytic spaces

Is it possible to define a Berkovich analytic space via its functor of points?
Let $k$ be a complete non-Archimedean field, possibly the trivial one. I am tempted to define a Berkovich analytic space ...

**7**

votes

**0**answers

348 views

### Etale cohomology of Berkovich spaces

Suppose $X/\mathbb{Q}$ is a reasonable smooth projective variety with interesting etale cohomology. For example, we can say $X$ is an elliptic curve. To what extent does it make sense to study the ...

**2**

votes

**1**answer

116 views

### Terminology: Epimorphism in non-archimedean analysis

In their book "Non-Archimedean analysis", when BGR refer to an epimorphism in the context of $k$-Banach algebras do they actually require (or does it follow that) such maps are surjections? For ...

**3**

votes

**1**answer

410 views

### How should we understand the relative interior in Berkovich spaces

I'm reading Berkovich's book on analytic spaces. The notion of relative interior confuses me. Is there anyway to see how it "looks like"? For instance, if $r <1$, what is the relative interior of
...

**3**

votes

**1**answer

238 views

### Spherical completions and flatness

Let $k$ be a non-Archimedean field. Does there exist a spherical completion $K$ of $k$ such that for any $k$-Banach space $X$, the natural map $X \to X \widehat{\otimes}K$ is an isometric embedding? ...

**12**

votes

**2**answers

1k views

### Rigid analytic spaces vs Berkovich spaces vs Formal schemes

I wonder if someone could explain briefly what is the relation between these 3 formal models, of a Berkovich space, a rigid analytic space and a formal scheme?
I have been working with formal schemes ...

**20**

votes

**1**answer

3k views

### Why is Faltings' “almost purity theorem” a purity theorem?

My understanding of purity theorems is that they come in several flavors:
1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob_p$ are algebraic numbers all of whose ...

**15**

votes

**1**answer

709 views

### Why do rigid spaces have “not enough points”?

In Brian Conrad's notes
here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero ...

**0**

votes

**0**answers

211 views

### morphism from adic spaces to schemes

Let $X:=Spa A$ be an affinoid adic space, and $\underline X $ the ringed space of $X$. Let $Y:=Spec B$ be an affine scheme, $f: \underline X \longrightarrow Y$ a morphism of ringed spaces.
How to ...

**3**

votes

**1**answer

185 views

### scheme of generalizations

Hi,
I have the following problem. Let $\mathcal{O}$ be a valuation ring and $S=Spec(\mathcal{O})$, denote with $s$ the closed point and with $\eta$ the generic one. Let $X\rightarrow S$ be a proper, ...

**11**

votes

**1**answer

538 views

### bornological vector spaces over a non-archimedean field

Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in ...

**5**

votes

**1**answer

594 views

### Reference Request: Vector bundles in rigid analytic geometry

In algebraic geometry it is well-known (see Hartshorne Exercise II.5.16 for example) that there is a 1-1 correspondence between rank $n$ (geometric) vector bundles $\pi\colon Y\to X$ on a scheme $X$ ...

**7**

votes

**1**answer

642 views

### Reference for rigid analytic GAGA

I'm looking for a reference for the following result.
Theorem. Let $K$ be a complete, non-archimedean field, and let $X/K$ be a projective scheme, with analytification $X^\mathrm{an}$. Then the ...

**12**

votes

**1**answer

587 views

### How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$?

I apologize in advance if this question is terribly naive. I've just recently learned a bit of rigid analytic geometry with the hopes of understanding some basic facts about eigenvarieties.
In the ...

**6**

votes

**0**answers

558 views

### Rigid Uniformization vs Grothendieck's Local Monodromy Theory

I've noticed that some interesting results about abelian varieties can each be proven using one of two ways: the theory of rigid uniformization of abelian varieties or Grothendieck's local monodromy ...

**6**

votes

**2**answers

444 views

### Uniqueness of analytic continuation in rigid analytic geometry

In classical complex analysis it is easy to prove that a meromorphic function has at most one analytic continuation (on an open connected subset of $\mathbb C$, say).
The problem of non-uniqueness of ...

**2**

votes

**1**answer

223 views

### Modules with connection over $p$-adic laurent series rings

If $X$ is a smooth rigid analytic space over a $p$-adic field $K$ (of characteristic zero), then every coherent $\mathcal{O}_X$-module with integrable connection is locally free. In his paper ...

**8**

votes

**1**answer

509 views

### Analytic elements in non-archimedean geometry

Let $(k,|.|)$ be a complete non-archimedean valued field. Let $D$ be the open unit disc over $k$. (Anything I write could be adapted to the case of an open annulus.) The ring $\mathcal{O}(D)$ of ...

**1**

vote

**0**answers

361 views

### Component group of Neron model of a parametrized abelian variety

Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an ...

**2**

votes

**0**answers

123 views

### Support of Tor over affinoid algebras

Suppose $k$ is a complete nonarchimedian field, $A$ is a $k$-affinoid algebra, and $M$ is a finitely presented $A$-module. Is the set
$\tau(M)= \left\{ x \in ...

**6**

votes

**1**answer

516 views

### $p$-adic uniformization not from the Drinfel'd spaces?

It seems that when we talk about the $p$-adic uniformization, we typically mean those uniformized by either the Drinfel'd upper spaces (for which we think of the examples of Mumford curves and some ...

**5**

votes

**1**answer

612 views

### Base Change for Eigenvarieties

Let $E/F$ be a Galois extension of number fields, and $G$ a reductive group over $F$. If Langlands Base Change is known for $G/F$ and $G/E$, and moreover the eigenvarieties for $G/F$ and $G/E$ have ...

**2**

votes

**0**answers

315 views

### reference for p-adic Stein spaces

Hi,
I'm looking for a reference in english for p-adic Stein spaces. The usual referneces I come across are all in german.
Thanks

**14**

votes

**1**answer

736 views

### D-modules on rigid analytic spaces

Is there a good notion of holonomic $D$-modules on rigid analytic spaces?

**3**

votes

**1**answer

302 views

### universal finite differential module of affinoid algebra

Let $k$ be a value field (archimedean), for example $k = \mathbb{Q}_p$, the p-adic field.
The free Tate algebra is $$ T_n := \left\{ \ \sum a_I X^I, \ a_I \in k, \ a_I \rightarrow 0 \text{ as } |I| ...

**10**

votes

**0**answers

405 views

### Is the Gouvea-Mazur problem related to symmetric square $L$-functions?

Here's an idea that I've found appealing but have never been able to get anywhere with.
One way to frame the Gouvea-Mazur question (for lack of a better term, since the original conjecture by the ...