Questions tagged [non-archimedean-fields]

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Does antidifferentiability of continuous functions imply Dedekind completeness?

Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...
James Propp's user avatar
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9 votes
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637 views

In need of help with parsing non-Archimedean function theory

My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I'...
MCS's user avatar
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7 votes
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283 views

Analogs of the Weil conjectures for non-archimedian fields

Suppose that $X$ is a smooth and proper variety defined over a perfect non-archimedian valued field $k$ of characteristic $p$.  Then one can consider the action of Frobenius on crystalline cohomology. ...
Laurent Cote's user avatar
7 votes
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308 views

Berkovich spaces — why use atlases?

Note: this question only concerns Berkovich spaces ("analytic spaces") as defined in Berkovich's first book "Spectral Theory and Analytic Geometry over Non-Archimedean Fields", not ...
Wojowu's user avatar
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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 ...
PrimeRibeyeDeal's user avatar
7 votes
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242 views

Singularities of an analytic function over a non-archimedean field

What do we know about the types of singularities that a convergent power series over a non-archimedean field can have? More specifically: i) What types of essential singularities can occur? ii) Are ...
Helene Sigloch's user avatar
6 votes
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291 views

Abelianization of the inertia group

Let $F/\mathbb Q_p$ be a finite extension, and let $I_F=\operatorname{Gal}(\overline F/F^{\mathrm{unr}})\subset\operatorname{Gal}(\overline F/F)$ be the inertia subgroup. Is there a description of ...
Kenta Suzuki's user avatar
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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}{...
AZZOUZ Tinhinane Amina's user avatar
5 votes
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137 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?
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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 :...
David Loeffler's user avatar
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203 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 ...
Chilote's user avatar
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Projective reduction of image of power series is algebraic?

Let $K$ be a non-archimedean field with closed unit disk $\mathcal{O}\subset K$, open unit disk $\mathfrak{m}\subset \mathcal{O}$ and residue field $k = \mathcal{O}/\mathfrak{m}$. Examples to keep in ...
Jef's user avatar
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What information does the topology of nonarchimedean Berkovich analytic spaces encode?

Given a finite type scheme $X$ over $\Bbb{C}$ we can associate to it an analytic space $X^\text{an}$. There are then comparison theorems comparing invariants of the topological space $X^\text{an}$ ...
Nuno Hultberg's user avatar
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137 views

Coherence of the I-adic completion of a local ring of a formal scheme

Let $K$ be a valued field of rank one and $K^+$ its valuation ring such that $K^+$ is $\varpi$-adically complete with respect to a pseudo-uniformizer $\varpi\in K^+$. Let $X$ be a smooth finite type $...
Takagi Benseki's user avatar
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109 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}}:=\...
Ale's user avatar
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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 $\{a_i\} \in D(a,r)$ ...
Panna's user avatar
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139 views

Interpreting umbral calculus in terms of some kind of extended numbers

I know that currently umbral calculus is developed as some kind of theory of operators and functionals but were there any attempts to put it on a more solid philosophical grounds as study of functions ...
Anixx's user avatar
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Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras

I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
Very Forgetful Functor's user avatar
3 votes
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180 views

$C^*$-algebras over an extension of $\mathbb{Q}_p$?

I'm wondering to what extent it might be possible for the theory of $C^*$-algebras to be translated into the $p$-adic context i.e. to define 'p-adic $C^*$-algebras' over some extension of $\mathbb{Q}...
Very Forgetful Functor's user avatar
3 votes
0 answers
169 views

Wildly ramified extension field

Given an algebraically closed complete valued field $(k,|.|)$ with characteristic 0, such that the residue field $\tilde{k}$ has a positive characteristic, and consider the complete extension $(\...
AZZOUZ Tinhinane Amina's user avatar
3 votes
0 answers
177 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)...
quasi-mathematician's user avatar
3 votes
0 answers
321 views

Asymptotics vs Puiseux series

Define asymptotic as a class of sequences {$ x_i$},$_{i\in\mathbb N}$ modulo equivalence {$x_i$}={$y_i$} if $\lim_{i\to\infty} (x_i/y_i)=c\in\mathbb R,c\ne 0$. More, we define $X= \{x_i\} \lt Y= \{ ...
Nikita Kalinin's user avatar
2 votes
0 answers
111 views

p-adic Banach space and complete tensor product

Let $p$ be a prime and $\mathbb{C}_{p}$ the completion of the algebraic closure of the $p$-adic number field $\mathbb{Q}_p$. Let $M$ be a $\mathbb{Q}_p$-Banach space. We denote by $M\mathbin{\widehat{\...
user521844's user avatar
2 votes
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211 views

Enlightening examples of tropical skeletons of Berkovich spaces

Let $K$ be a complete non-archimedean field and let $X$ be a $K$-analytic space in the sense of Berkovich of pure dimension $d$. Let $\varphi \colon X \to \mathbf{G}_m^r$ be a moment map to an ...
user avatar
2 votes
0 answers
72 views

Filtration of norm-one elements of quaternion algebra over local field with respect to an involution

Let $K$ be a local non-archimedean field, with ring of integers $\mathcal{O}_K$, uniformizing element $\varpi_K$, and residue field $\mathcal{O}_K/\varpi_K\mathcal{O}_K \cong \mathbb{F}_q$. For ...
pbarron's user avatar
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179 views

A generalisation of closed and bounded subsets of non-Archimedean fields to topological spaces

The definition of compactness in topological spaces generalises the notion of a subset of $\mathbb{R}^n$ being closed and bounded, as expressed by the Heine-Borel Theorem. In finite-dimensional vector ...
Very Forgetful Functor's user avatar
2 votes
0 answers
130 views

Composite of two fields contain a given quadratic extension, but each individual doesn't

In fact, this question could be asked for arbitrary field extension. However, for simplicity I only ask the question for local field of characteristic 0. Let $E/F$ be a quadratic extension of padic ...
gg123's user avatar
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0 answers
241 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-...
user avatar
2 votes
0 answers
112 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 ...
MCS's user avatar
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2 votes
0 answers
67 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 ...
Kiu's user avatar
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2 votes
0 answers
167 views

Projective tensor powers of Banach spaces over a normed field

Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
Vladimir's user avatar
  • 359
1 vote
0 answers
57 views

A certain subalgebra of $\mathfrak{e}_8$ over a p-adic field

Can $\mathfrak{e}_8(\mathrm{k})$ have a maximal subalgebra isomorphic to $\mathfrak{sl}_1(\mathrm{D})\oplus\mathfrak{g}_2(\mathrm{K})$, where $\mathrm{k}$ is a finite extension of some $\mathbb{Q}_p$, ...
Daniel Sebald's user avatar
1 vote
0 answers
98 views

Does maximally incompleteness cause nonvanishing of the extension of maximal ideal of a valuation ring by rank 1 free module?

In B. Bhatt's lecture notes[1], Remark 4.2.5 says ... $\operatorname{Ext}_R^2(k,R)$ is non-zero if $K$ is not spherically complete. which amounts to the following pure algebraic question. Statement ...
XYC's user avatar
  • 389
1 vote
0 answers
177 views

Contractibility of the quotient of an analytification of a smooth variety by a finite group (if the field is trivially valued)

Let $k$ be a field and $X$ be a smooth irreducible $k$-variety with an action of a finite group $G$. I consider $k$ as a trivially valued field. It is known from results of Berkovich ("Smooth p-...
Sam's user avatar
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1 vote
0 answers
187 views

Tate uniformization and reduction of elliptic curves

Let $E$ be an elliptic curve over $K$ (nonarchimedean) with $j$-invariant satisfying $|j(E)|>1$. Tate uniformization theorem says that we have an isomorphism : $E \simeq \mathbf G_m/q^{\mathbf Z}$. ...
zodiack's user avatar
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1 vote
0 answers
108 views

What is some algebraic intuition behind the fact that the (real part) of the logarithm of Bernoulli umbra plus $1$, is $-\gamma$?

Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus. Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\...
Anixx's user avatar
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1 vote
0 answers
480 views

'Spherically complete' normed fields

A non-Archimedean normed field $K$ is said to be spherically complete if every decreasing sequence of closed balls in $K$ has non-empty intersection. I am a little puzzled as to why this definition is ...
Very Forgetful Functor's user avatar
1 vote
0 answers
104 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 $\...
joaopa's user avatar
  • 3,655
1 vote
0 answers
220 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 ...
user avatar
1 vote
0 answers
68 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 ...
Hang's user avatar
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1 vote
0 answers
162 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 $(...
joaopa's user avatar
  • 3,655
1 vote
0 answers
129 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 $f_i\in T_n$....
Hang's user avatar
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1 vote
0 answers
241 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-...
Hang's user avatar
  • 2,719
1 vote
0 answers
143 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 ...
Dima Sustretov's user avatar
1 vote
0 answers
112 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 ...
Justin Campbell's user avatar
0 votes
0 answers
67 views

Space of non-archimedean characters is nonempty

Let $k$ be an algebraically closed complete non-archimedean field. Let $\mathcal{O}_k$ be its ring of integers. Suppose that $A$ is a $k$-Banach algebra, and $B$ is its closed unitary ball. Note that $...
Luiz Felipe Garcia's user avatar
0 votes
0 answers
86 views

Terminology for discrete subgroups of PSL(2,k), where k is a non-archimedean local field

$\DeclareMathOperator\PSL{PSL}$I'm asking about terminology for discrete subgroups of $\PSL(2,k)$, where $k$ is a non-archimedean local field. As it is rather clumsy to have to use such expressions ...
Hercule Poirot's user avatar