**26**

votes

**1**answer

2k views

### Structure on $X(k)$ for separated finite type alg. space $X$, for complete valued $k$.

Let $k$ be a field complete with respect to a non-archimedean absolute value, and $X$ a separated algebraic space of finite type over $k$.
If $X$ is a scheme then $X(k)$ inherits a natural ...

**18**

votes

**0**answers

508 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 ...

**17**

votes

**1**answer

2k 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 ...

**14**

votes

**1**answer

674 views

### D-modules on rigid analytic spaces

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

**14**

votes

**1**answer

618 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 ...

**13**

votes

**1**answer

908 views

### Are flat morphisms of analytic spaces open?

Let $f:X\to Y$ be a morphism of complex analytic spaces. Assume $f$ is flat (or, more generally, that there is a coherent sheaf on $X$ with support $X$ which is $f$-flat). Is $f$ an open map?
The ...

**11**

votes

**2**answers

905 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 ...

**11**

votes

**1**answer

487 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 ...

**11**

votes

**1**answer

508 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 ...

**11**

votes

**0**answers

507 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 ...

**10**

votes

**1**answer

598 views

### Consequences of the geometric properties of the eigencurve

The eigencurve $\mathcal{E}$ is a rigid-analytic space parametrizing certain $p$-adic families of modular forms and associated Galois representations. By constructing an auxiliary reduced rigid curve ...

**10**

votes

**2**answers

425 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 ...

**10**

votes

**0**answers

385 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 ...

**8**

votes

**2**answers

946 views

### Cohomology of rigid-analytic spaces

Let $R$ be a complete discrete valuation ring and let $K$ be its field of fractions. Suppose $X$ is a smooth rigid-anaytic space over $K$. Often it is convenient to have a model of $X$ whose ...

**8**

votes

**1**answer

418 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 ...

**8**

votes

**1**answer

970 views

### Do Berkovich homogenous spaces exist?

Let G be a k-analytic group, and let H be a closed subgroup of G. Then does there exist a k-analytic space, which can be reasonably called the quotient G/H?
Commentary: I realise that I am not being ...

**7**

votes

**1**answer

506 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 ...

**6**

votes

**3**answers

424 views

### Weierstrass points on rigid-analytic surfaces

Does a rigid-analytic surface defined over a nonarchimedean complete field have Weierstrass points (if its genus is big enough let's say)? Is there a good reference that (ideally) lists theorems for ...

**6**

votes

**1**answer

533 views

### Tate models for semistable algebraic varieties with mixed reduction over a local field

It's known that if $A$ is an abelian variety of totally multiplicative reduction over a p-adic field K, then, after taking a finite field extension, it becomes isomorphic, as a rigid analytic group, ...

**6**

votes

**1**answer

297 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 ...

**6**

votes

**2**answers

401 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 ...

**6**

votes

**1**answer

433 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 ...

**6**

votes

**0**answers

116 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? ...

**6**

votes

**0**answers

293 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 ...

**6**

votes

**0**answers

510 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 ...

**5**

votes

**1**answer

573 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 ...

**5**

votes

**1**answer

157 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 $k$.
...

**4**

votes

**1**answer

254 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 $\mathbb{Q}_p$.
...

**4**

votes

**1**answer

529 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 $X$ ...

**4**

votes

**0**answers

710 views

### Generalized GAGA

So, I have heard GAGA works for Rigid Analytic spaces. I know next to nothing about this, but it made me curious as to whether there are any other contexts in which GAGA "works". Of course, this is a ...

**3**

votes

**1**answer

360 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

211 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? ...

**3**

votes

**1**answer

173 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, ...

**3**

votes

**1**answer

279 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| ...

**3**

votes

**0**answers

462 views

### What is the nature of the zero locus of a section of a coherent sheaf?

Suppose that $X$ is a reduced rigid space and $\scr{F}$ is a coherent sheaf on $X$. For a section $f\in {\scr F}(X)$, the zero locus of $f$ is the set of points $x\in X$ at which $f$ vanishes in the ...

**2**

votes

**1**answer

130 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 ...

**2**

votes

**1**answer

108 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 ...

**2**

votes

**1**answer

187 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 ...

**2**

votes

**1**answer

199 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 ...

**2**

votes

**0**answers

119 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 ...

**2**

votes

**0**answers

291 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

**2**

votes

**0**answers

287 views

### Sheaf of power-bounded elements in rigid analytic geometry

Let $k$ be a field with a non-archimedean complete valuation $|\ |$, $X$ a reduced rigid analytic space over $k$. The presheaf $\mathcal{O}^0$ which to an affinoid $U$ of $X$ attaches the ring ...

**1**

vote

**1**answer

444 views

### What are in units of an affinoid algebra?

Suppose that $K$ is a complete local field and $A$ is an affinoid $K$-algebra. Is there a known way to produce an explicit description of the units of $A$?
Here is what I already know: write ...

**1**

vote

**0**answers

49 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 ...

**1**

vote

**0**answers

306 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 ...

**0**

votes

**0**answers

70 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 ...

**0**

votes

**0**answers

39 views

### closed subscheme associated to a closed rigid subspace

Let me start by saying that I know almost nothing about rigid geometry and I have no idea whether my question is easy or not.
Start with a ring $R$ which is local, complete for the topology defined ...

**0**

votes

**0**answers

189 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 ...