Questions tagged [non-archimedean-fields]
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47
questions with no upvoted or accepted answers
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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 ...
9
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637
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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'...
7
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283
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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. ...
7
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308
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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 ...
7
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431
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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 ...
7
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242
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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 ...
6
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291
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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 ...
6
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145
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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}{...
5
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137
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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?
5
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363
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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 :...
5
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203
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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 ...
4
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107
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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 ...
4
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0
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219
views
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}$ ...
4
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137
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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 $...
4
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109
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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}}:=\...
4
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145
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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)$ ...
3
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139
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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 ...
3
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145
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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 ...
3
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0
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180
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$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}...
3
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169
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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 $(\...
3
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177
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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
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321
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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= \{ ...
2
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111
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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{\...
2
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211
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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 ...
2
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72
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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 ...
2
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179
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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 ...
2
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0
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130
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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 ...
2
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241
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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-...
2
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112
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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 ...
2
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0
answers
67
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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 ...
2
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167
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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 ...
1
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57
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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$, ...
1
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98
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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 ...
1
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177
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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-...
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0
answers
187
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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}$.
...
1
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0
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108
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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 $\...
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0
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480
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'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 ...
1
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0
answers
104
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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
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0
answers
220
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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 ...
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68
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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 ...
1
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answers
162
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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 $(...
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129
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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 $f_i\in T_n$....
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0
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241
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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-...
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143
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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 ...
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112
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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 ...
0
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0
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67
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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 $...
0
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86
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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 ...