Questions tagged [non-archimedean-fields]

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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. ...
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4 votes
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An example where the non-Archimedean tensor product of normed modules is only seminormed?

Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}_{\geq 0}$ which satisfies $$ | 0_R| = ...
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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 ...
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1 vote
1 answer
138 views

Reference request: Gruson's theorem on the tensor product of Banach spaces over a non-Archimedean field

I am looking for a reference for theorem 3.21 of these notes: https://web.math.princeton.edu/~takumim/Berkovich.pdf The theorem states that if $k$ is a non-Archimedean field and $X$ and $Y$ are $k$-...
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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 ...
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Levi-Civita analog of imaginary unit?

If we consider the following element of Levi-Civita field: $I=\frac{\sqrt{1-\varepsilon ^2}}{\varepsilon }$ then its consecutive powers would have the property $\operatorname{st}I^n=\operatorname{Re}(...
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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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3 votes
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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 ...
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8 votes
2 answers
354 views

Literature on non-Archimedean analogues of basic complex analysis results

It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions and ...
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Non-Archimedean Lebesgue dominated convergence theorem

In this paper, the authors explain that the full generality of the Lebesgue dominated convergence theorem holds for functions on a compact zero-dimensional space $X$ taking values in a metrically ...
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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 ...
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2 votes
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126 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 ...
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9 votes
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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'...
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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}...
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7 votes
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254 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 ...
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1 answer
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Product absolute value in rings of integers

Let $F$ be an algebraically closed field of characteristic $p$ equipped with a nonarchimedean dense absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete. Let ...
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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 ...
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2 votes
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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 ...
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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 $(\...
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2 votes
1 answer
118 views

Reference to basic facts on non-Archimedean local fields

I need a reference to the following claims which, I believe, are correct and well known to experts (I am not one of them). Let $K$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of ...
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3 votes
1 answer
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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 ...
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37 votes
2 answers
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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-...
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7 votes
1 answer
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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 ...
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2 votes
1 answer
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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 $\...
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1 vote
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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 $\...
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  • 3,079
1 vote
1 answer
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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 ...
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3 votes
1 answer
250 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 ...
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5 votes
0 answers
106 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}{...
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7 votes
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284 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 ...
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1 vote
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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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3 votes
0 answers
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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)...
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3 votes
1 answer
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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 ...
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9 votes
1 answer
661 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 ...
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4 votes
2 answers
292 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)\...
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2 votes
0 answers
199 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-...
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5 votes
0 answers
135 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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3 votes
1 answer
100 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 ...
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  • 19k
1 vote
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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 ...
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5 votes
0 answers
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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 :...
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4 votes
1 answer
181 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 ...
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2 votes
0 answers
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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 ...
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  • 1,216
1 vote
0 answers
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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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  • 3,079
1 vote
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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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  • 2,611
1 vote
0 answers
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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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  • 2,611
3 votes
1 answer
379 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 $f\in T_n$...
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  • 2,611
2 votes
1 answer
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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....
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  • 155
3 votes
1 answer
240 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/...
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  • 155
8 votes
1 answer
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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 ...
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  • 2,611
7 votes
1 answer
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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 ...
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  • 2,611
13 votes
2 answers
815 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 ...
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