# Questions tagged [non-archimedean-fields]

The non-archimedean-fields tag has no usage guidance.

71
questions

**3**

votes

**1**answer

276 views

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

I am reading the following article of Berger, p8 and I don't understand the idea:
$C_p:=\widehat{\overline{\mathbb Q_p}}$ does not contain the periods
The text seem to reason as follows
(under some ...

**32**

votes

**2**answers

2k views

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

Recently in a seminar the following question was raised and, despite my familiarity with theory, I couldn't come up with a good answer:
Are there any good reasons to use Tate's theory of rigid-...

**7**

votes

**1**answer

225 views

### Hahn’s theorem on ordered fields

There is a theorem attributed to Hahn that every ordered field $F$ containing $\mathbb R$ is a subfield of a formal power series field $\mathbb R[[X^\Gamma]]$, where $\Gamma$ is an ordered abelian ...

**2**

votes

**1**answer

168 views

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

Let $k$ be a non-archimedean field and denote by $\mathbb{A}_k^n$ the analytic affine space of $n$ dimensions over $k$ (analytic in the sense of Berkovich). There is a natural injective map of sets $\...

**1**

vote

**0**answers

98 views

### Algebraic morphisms of affine varieties in positive characteristic

Let $\Omega$ be a completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the valuation $-\deg$.
Consider two matrices $M_1,M_2$ in $\mathcal M_2(\Omega)$ that are $\...

**1**

vote

**1**answer

119 views

### Theory of extensions of non-archimedian local fields

I'm searching for a recommendable reference dealing with theory of
non-Archimedean local fields where I can find proofs of the following claims about
finite extensions $L/K$ of non-Archimedean local ...

**2**

votes

**1**answer

214 views

### perfectoid field of characteristic $p$

Let $L$ be perfectoid field of characteristic $p$ and $L'$ be a finite extension of $L$. Then I want to prove the trace map $\text{Tr}_{L'/L}: m_{L'}\rightarrow m_L$ is surjective. I find a proof in ...

**5**

votes

**0**answers

82 views

### The valuation of finite extension of an non-archimedean field

Let $(k,|.|)$ be a non-archimedean complete field and $(L,|.|)$ be a finite extension of $(k,|.|)$, $[L:k]=n$, such that $L=k(\xi)$. Let $\phi$ the homomorphism of $k$-Banach algebra
$$\begin{array}{...

**6**

votes

**0**answers

203 views

### intuition for lattices in p-adic (or other non-Archimedean) vector spaces?

I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$.
I have some intuition for $\mathbb{Z}$-lattices ...

**1**

vote

**0**answers

100 views

### compact $p$-adic Lie group can be embedded into $O_K^n$ or $\text{GL}_n(K)$?

Let $K$ be a local field of charecteristic $0$ and $G$ be a compact $p$-adic Lie group of dimension $n$, then can $G$ be embedded into $O_K^n$ or $\text{GL}_n(K)$ as a closed subgroup? This is a dual ...

**3**

votes

**0**answers

141 views

### gluing Berkovich spaces

In his paper Etale cohomology for non-Archimedean analytic space (IHES), Berkovich explained how to glue $k$-analytic spaces (Prop. 1.3.3) and show its uniqueness using the Prop 1.3.2 (gluing morphism)...

**3**

votes

**1**answer

133 views

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

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $V$ be a Banach algebra over $K$, then what is the $K$-analytic space corresponding to $V$? What is the definition of $K$-analytic space? This is ...

**9**

votes

**1**answer

534 views

### Completed tensor product is exact

In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on ...

**4**

votes

**2**answers

285 views

### $p$-adic series bounded if and only if it has finitely many zeros

Let $L\subseteq\mathbb{C}_p$ be a finite extension of $\mathbb{Q}_p$, $r$ be a positive real number, and $f$ be a series $\sum_{n\in \mathbb{Z}} a_nz^n$ convergent in $D= \{x\in \mathbb{C}_p|0<v(x)\...

**2**

votes

**0**answers

173 views

### Rigid analytic geometry and Tate curve

I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-...

**5**

votes

**0**answers

132 views

### Fields that are not finite extensions of proper subfields

What fields are not finite extensions of proper subfields? Prime fields and (less obviously) real closed fields have this property. Do the $p$-adics enjoy this property as well?

**3**

votes

**1**answer

98 views

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

EDIT Let $\mathcal{O}$ be the ring of integers in a non-Archimedean local field. Let $GL_n(\mathcal{O})$ be the (compact) group of $n\times n$ matrices with entries in $\mathcal{O}$ such that its ...

**1**

vote

**0**answers

54 views

### Polytopal domains in non-archimedean torus

Given a non-archimedean field $\mathbb K$, there is a natural map
$$
\mathrm{val}: (\mathbb K^*)^n\to\mathbb R^n$$
(See Section 4 of Gubler's paper).
Gubler mentions there $\mathrm{val}$ is a ...

**5**

votes

**0**answers

196 views

### Is this subset of a rigid space an admissible open?

Let $K$ be a $p$-adic field and let $X$ be the rigid space $ \operatorname{Max} K\langle T_1, T_2 \rangle$, i.e. the 2-dimensional closed unit ball.
Consider the sets $U := \{ |T_1| < 1\}$ and $V :...

**4**

votes

**1**answer

164 views

### Significance of integrally closed in an affinoid algebra

A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.
See for ...

**2**

votes

**0**answers

72 views

### Calculus over Function Fields of Characteristic Zero

Having done some cursory searching of the internet, it is clear to me that there is a very well-developed theory of how to do calculus over function fields, such as fields of Laurent series in a ...

**1**

vote

**0**answers

148 views

### Automorphisms of $\mathbb C_p$ with constraints

In Automorphisms of $\mathbb C_p$, K. Conrad showed that there exist uncountably many $\mathbb Q_p$-automorphisms of $\mathbb C_p$. I have a quite similar question.
Let $(a_n)_{n\in\mathbb N}$ and $(...

**1**

vote

**0**answers

106 views

### Affinoid algebra and fundamental theorem of algebra

This post is closely related to the previous one here.
But more generally, we want to study an affinoid algebra $A:=T_n/\mathfrak a$. Let's assume $\mathfrak a= (f_1,\dots,f_r)$ for some $f_i\in T_n$....

**1**

vote

**0**answers

141 views

### Explicit description of rigid analytification of torus

It is known that in non-archimedean world there is also a GAGA-functor from the category of $K$-schemes of locally finite type to the category of rigid $K$-spaces. Here $K$ is a field with a non-...

**3**

votes

**1**answer

342 views

### Tate algebras and fundamental theorem of algebra

Let $\mathbb K$ be an algebraically-closed complete non-archimedean field whose absolute value is non-trivial. Consider the Tate algebra $T_n=\mathbb K\langle X_1,\dots, X_n \rangle$ and fix $f\in T_n$...

**2**

votes

**1**answer

123 views

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

I am (still) working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10....

**3**

votes

**1**answer

201 views

### Definition of a vertical ideal sheaf and a vertical fractional ideal sheaf

I'm working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10.1090/jag/...

**8**

votes

**1**answer

353 views

### Raynaud's universal Tate elliptic curves

In the end of Section 9.2 of Bosch's book Lectures on Formal and Rigid Geometry, a rigid $S$-space $E_Q$ is constructed, for a variable $Q$ replacing the classical parameter $q\in k$. (Here $k$ is a ...

**7**

votes

**1**answer

308 views

### Formal power series in Berkovich geometry

In this Berkovich's paper, the following kind of algebra is studied:
$$
A=A_{m,n}=k^\circ \langle T_1,\dots,T_m \rangle [[S_1,\dots,S_n]]
$$
where $k$ is some non-archimedean field with non-trivial ...

**13**

votes

**2**answers

717 views

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

From this Temkin's paper (at the end of section 1.1.3), I know that one may define Berkovich spaces that include both archimedean and non-archimedean worlds. This looks very interesting.
Temkin ...

**4**

votes

**1**answer

354 views

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

Chambert-Loir and Ducros have introduced real differential forms and currents on Berkovich spaces.(See Gubler's survey for example). In that survey, a tropical chart $V$ is defined on an ...

**4**

votes

**0**answers

96 views

### On properties of affinoid algebras preserved under reduction

Let $R$ be a complete valuation ring of rank one, with maximal ideal $m$ and residue field $k$. Consider a $K$-affinoid algebra $\mathcal{A}$ and its reduction given by $\widetilde{\mathcal{A}}:=\...

**0**

votes

**1**answer

68 views

### Discrete valuation sub-fields

If $C$ is a characteristic $0$ algebraically closed field over $\mathbb{Q}_p$ which is complete with respect to a non-trivial non-archimedean valuation. Now let $x_1, \cdots,x_n$ be any $n$ elements ...

**6**

votes

**1**answer

222 views

### Is any nonarchimedean field containing all roots of unity perfectoid?

Say $K$ is a complete nonarchimedean extension of $\mathbf{Q}_p$, i.e., it is the fraction field of a $p$-adically complete and $p$-torsionfree rank $1$ valuation ring. Assume that the residue field ...

**1**

vote

**1**answer

353 views

### Similarity and difference between Archimedian and non-Archimedian geometry [closed]

This question is on similarity and difference between Archimedian and non-Archimedian field.
An example of non-Archimedian field is the field of $p$-adic numbers $\mathbb{Q}_p$.
We have concept of ...

**7**

votes

**2**answers

291 views

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

My question is rather simple. Do there exist a formally real field that admits a unique ordering (so sums of squares are the positive elements) and such that this ordering is not archimedean?
Oh, I ...

**8**

votes

**2**answers

592 views

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

Fontaine, J.-M.; Illusie, L.
p-adic periods: A survey. (English)
Ramanan, S. (ed.) et al., Proceedings of the Indo-French conference on geometry held in Bombay, India, 1989.
If anyone has the above ...

**4**

votes

**1**answer

97 views

### finite number of vertices of the polyhedron of variation of an invertible function on a Berkovich curve

The paper of Ducros "Cohomologie non-ramifiée sur une courbe p-adique lisse" mentions a theorem (1.21) about the existence of a "polyhedron of variation" of an invertible function on a Berkovich curve....

**2**

votes

**0**answers

53 views

### Terminology for valuation-like functions on a vector space

Let $V$ be a vector space over a field $k$. I was wondering if there is a standard terminology for a function $v: V \setminus \{0\} \to \mathbb{R}$ which is invariant under multiplication by nonzero ...

**3**

votes

**2**answers

327 views

### generic fibre functor for relative rigid spaces

The classical theory of formal models of rigid analytic spaces due to
Raynaud introduces the category of admissible R-formal schemes for $R$ a
discretely valued ring, which includes locally ...

**4**

votes

**0**answers

135 views

### The image of annuli of the non-Archimedean projective line by rational functions

I'm reading the book "potential theory and dynamics over the Berkovich projective line" by Baker and Rumely. The proposition 2.18 in this claims that if you choose suitable finite $\{a_i\} \in D(a,r)$ ...

**1**

vote

**0**answers

90 views

### characterisation of definable functions in the Lipschitz-Robinson expansion

A subset $X$ of a domain $\Omega$ in a real analytic manifold $M$ is called semi-analytic if in a neighbourhood of any point $p \in M$ it is described by finitely many equations and inequalities in ...

**1**

vote

**0**answers

130 views

### characterization of the subspace of the moduli space of curves with maximally degenerate Jacobian

Let $K$ be a field equipped with a non-Archimedean absolute value, for example $K=\mathbb{C}((t))$. An Abelian variety $A$ over $K$ is called maximally degenerate if it admits an analytic ...

**5**

votes

**0**answers

171 views

### Completions of $K(x)$

Let $K$ be a field. Are there books or articles discussing completions of $K(x)$ with respect to the metric induced by the $p$-adic valuation $|\;\;|_p$ where $p\in k[x]$ is irreducible and different ...

**3**

votes

**1**answer

78 views

### the structure on the value group sort of a C-minimal field

Let $K$ be an algebraically closed valued field which is $C$-minimal, as defined, for example, in this article. Examples include pure algebraically closed valued fields, as well as Lipschitz and ...

**5**

votes

**1**answer

152 views

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

If $F$ is an ordered field and $G$ is an ordered abelian group, one can define the Hahn product $F \boxtimes G$ to be the set of formal Laurent series with coefficients in $F$ and exponents in $G$. It ...

**6**

votes

**1**answer

345 views

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

I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature.
Let us fix a non-archimedean complete valuation field $...

**6**

votes

**1**answer

299 views

### Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field.
Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that:
$||x||=0$ if and only if $x=0$,
$||\lambda x||=|...

**2**

votes

**2**answers

476 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

**1**answer

391 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, ...