31
votes

### Are rigid-analytic spaces obsolete, since adic spaces exist?

My opinion (knowing that this is somehow a matter of taste) is that the rigid-analytic viewpoint is more or less obsolete, and that one should better use Berkovich, Huber or Raynaud instead (each of ...

30
votes

Accepted

### Are rigid-analytic spaces obsolete, since adic spaces exist?

There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications?
For the first question, it'...

20
votes

Accepted

### Berkovich space including both archimedean and non-archimedean worlds

The definition of analytic space over $\mathbf{Z}$ was given by Berkovich in his foundational book "Spectral theory and analytic geometry over non-Archimedean fields" (see the beginning of section 1.4 ...

12
votes

### Mixing solids and liquids

Good question!
I think the real context for the question was whether certain objects that are implicit in work of Darmon (and collaborators) could exist within this framework of analytic geometry. ...

11
votes

Accepted

### Hahn’s theorem on ordered fields

This theorem, which extends Hahn's embedding theorem for ordered abelian groups to ordered fields, has a complicated history that makes it difficult to attribute it to any single author. However, by ...

10
votes

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

Yes, there is a good notion of dimension, due to Berkovich and developed in my article, as mentioned in the two answers above.
Concerning your question about GAGA principle for pure dimension, the ...

9
votes

### Berthelot functor, rigid analytic space

The setup of the question is not general enough:
(i) you mean to work with Spf rather than Spec,
(ii) Raynaud's construction doesn't apply to the formal scheme Spf($A$) for such $A$,
(iii) the ...

Community wiki

9
votes

Accepted

### Why does $\mathbb C_p$ not contain the periods?

Consider the Tate motive $\mathbb Q(1)$. Its de Rham realization is simply $\mathbb Q$ (with the filtration $F^{-1}\mathbb Q=\mathbb Q$ and $F^{0}=0$) and its Betti realization is $2\pi i\mathbb Q$. ...

7
votes

Accepted

### Literature on non-Archimedean analogues of basic complex analysis results

Benedetto has a textbook that discusses basic $p$-adic analysis, although his aim is to study $p$-adic dynamics. And it's for a single variable. But might be a good place to get some information.
...

6
votes

Accepted

### Fontaine, J.-M.; Illusie, L. p-adic periods------Does any one have the following article?

It seems clear that there is no online version available, but a library can probably get a PDF copy through interlibrary loan if that's an option for you. The detailed listing is here. Note too ...

6
votes

### Formally real fields with unique non-Archimedean ordering

Yes, such fields exist. Let $T$ be the first-order theory in the language of fields with an extra constant $c$, axiomatized by
the axioms of fields,
every element or its negative is a sum of 4 ...

6
votes

Accepted

### Is there an exponential map on (Hahn) ordered fields?

There is no such exponential map. This was demonstrated in:
F.-V. Kuhlmann, S. Kuhlmann, S. Shelah, Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997) 3177–3183.

6
votes

Accepted

### How does an analytic space correspond to a $p$-adic Banach space

Without reading much beyond the pages adjacent to page 6, but looking at [ST3] (especially seeing the references [BGR] and [NFA]), I believe Berger-Colmez are using $K_n$-analytic spaces in the sense ...

6
votes

Accepted

### A question on linear algebra over non-Archimedean local field

$\def\FF{\mathbb{F}}\def\GL{\text{GL}}$This is basically the same thing YCor sketches in his comment: Let $R$ be the ring of integers of $\FF$. Let $K$ be the group of invertible matrices with entries ...

6
votes

Accepted

### Formal series which are always zero

Yes (assuming $k$ infinite: $k$ finite yields easy counterexamples), by a simple argument of elementary analysis.
Case $k$ infinite, discrete: then $f$ is a polynomial vanishing on $k^n$, hence is ...

6
votes

### Non-trivial extension of representations have same central character

You probably should assume all representations are smooth.
You hope to show if $\pi_1$ and $\pi_2$ have two different central characters $\phi_1$ and $\phi_2$ then any extension $0\to \pi_1\to \pi\to \...

6
votes

### Partition of unity for analytic manifolds over non-Archimedean local fields

It follows from Lemma 1 (part (2)) on page 7 in http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bernst_Lecture_p-adic_repr.pdf
In the non-compact case, I think that you need to ...

5
votes

Accepted

### Image of a finite dimensional complex representations of $GL_n(\mathcal{O})$

Call the representation $\pi$. Let $U$ be a neighbourhood of the identity in $\operatorname{GL}(V)$ that contains no non-trivial subgroup. Then the pre-image of $U$ under $\pi$ is a neighbourhood of ...

5
votes

Accepted

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

Resolution of singularities for rigid analytic varieties of equal characteristic zero follows from resolution of singularities for schemes of characteristic zero (Nicaise, A trace formula for rigid ...

5
votes

Accepted

### Relations between two definitions of non-archimedean analytic spaces

Let me give more a few more details than in nfdc23's comment. The most general definition of a analytic space is the one that you find in Berkovich's IHES paper. One requirement is that for every ...

5
votes

### Non-Archimedean non-standard models for R

Take for $S$ the field $F(t)$ with $F$ being the algebraic closure of $\mathbb Q$ inside $\mathbb R$. Equip it with the unique ordering for which $t-x>0$ for every integer $x$, and take a maximal ...

5
votes

Accepted

### Is $\mathbb{A}_k^n(k)$ dense in the Berkovich analytification of $\mathbb{A}_k^n$?

If $k$ is not algebraically closed, then $\mathbb A^n_k(k)$ is not doing to be dense with $\mathbb A_k^n$. I'll show this for $n=1$ for simplicity. Take any point $P$ in $\mathbb A^1_k$ with a residue ...

5
votes

Accepted

### Complete residue field of a point of type 5

A point of type 5 corresponds to a valuation of rank 2, so I am not sure if the meaning of completion is completely clear here. I think that the usual thing is to complete with respect to the ...

5
votes

Accepted

### Non-trivial extension of representations have same central character

@KentaSuzuki's answer is probably the best way of thinking about it, but here's another argument in line with what you wanted.
Permit me to write $\phi_1$ rather than $\phi$ for the central character ...

4
votes

### Fontaine, J.-M.; Illusie, L. p-adic periods------Does any one have the following article?

The desired paper is freely available on Fontaine's webpage! Here is the first part, and here is the second.

4
votes

Accepted

### Significance of integrally closed in an affinoid algebra

The adic spectrum of $(R,R^+)$ is a set of continuous valuations on $R$ having norm $\leq 1$ on $R^+$. It is clear from the definition and basic properties of integral closures that relaxing the ...

4
votes

### Berkovich space including both archimedean and non-archimedean worlds

You should take a look at the paper by Jérôme Poineau:
La droite de Berkovich sur Z, Astérisque n° 334 (2010)
and other papers by the same autor, like
Espaces de Berkovich sur Z : étude locale, ...

4
votes

Accepted

### Tropical charts (coordinates) and differential forms in non-archimedean geometry

I will try to explain what it is a tropical chart on an algebraic variety over a non-archimedean field $K$ (complete with respect to a non-archimedean absolute value, algebraically closed by ...

3
votes

Accepted

### Raynaud's universal Tate elliptic curves

Q1: Michel Raynaud, Géométrie analytique rigide d’après Tate, Kiehl..., Bull. Soc. Math. France, Mémoire 39-40, p. 319-327 (1974).

3
votes

Accepted

### Definition of model functions and their density in $C^0(X^\text{an})$

As alluded to in the question, the space $\mathcal{D}(X)_{\mathbb{Z}}$ is indeed the space of model functions arising from integral divisors.
Now, in order to deduce that $\mathcal{D}(X)_{\mathbb{Q}}$ ...

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