Questions tagged [non-archimedean-fields]
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130 questions
1 vote
0 answers
43 views
What are these ordered rings: for every $\epsilon>0$ there is an $\omega$ with $\omega\epsilon\ge 1$?
In my project I work with an ordered ring $R$ that has the following property: for all $\epsilon\in R$ with $\epsilon>0$ there is an $\omega\in R$ so that $\omega\epsilon\ge 1$. I wonder whether ...
- 14.5k
6 votes
0 answers
196 views
Can local metaplectic group exist as an analytic object?
Does there exist a group-object $Mp^{an}$ in the category of rigid/Berkovich/adic spaces over a non-archimedean field $k$, with a homomorphism $Mp^{an}\to Sp^{an}$ to the analytification of the ...
- 61
5 votes
0 answers
246 views
Non-Archimedean disks
Let $K$ be a field complete with respect to a non-Archimedean absolute value $|\cdot|$. To develop analysis in $K$, we need the notion of a disk in $K$. There is nothing mysterious at first glance: ...
- 1,354
14 votes
1 answer
452 views
Explicit witness to spherical incompleteness of $\mathbb{C}_p$
A nonarchimedean valued field $K$ is said to be spherically complete if, for any nested sequence $B_1 \supseteq B_2 \supseteq \dots$ of balls in $K$, the intersection $\bigcap_{i = 1}^\infty B_i$ is ...
- 283
0 votes
0 answers
52 views
Isometric map of affinoid p-adic algebras
Let $A = \mathbb{Q}_p\langle t_1, \dots, t_n \rangle = \mathbb{Q}_p\langle T_1, \dots, T_n \rangle/J$ be an $p$-adic affinoid algebra generated by $t_1, \dots, t_n$ with its norm being the quotient ...
- 1,607
2 votes
0 answers
79 views
Orthogonalization of quadratic forms over a $p$-adic Banach space
Let $X$ be an arbitrary set. Let $H = c_0(X, \mathbb{Q}_p)$ be the $p$-adic Banach space with sup norm. Let $\langle \cdot, \cdot \rangle$ be a symmetric, nondegenerate $\mathbb{Q}_p$-bilinear form on ...
- 1,607
1 vote
0 answers
69 views
Open solvable subgroups of non-Archimedean Lie groups
Let $K$ be a local non-Archimedean field and $G$ a closed nondiscrete subgroup of $\mathrm{GL}_n(K)$, or a $K$-Lie analytic group (though I am primarily interested in linear groups). Is the following ...
- 11
3 votes
1 answer
248 views
Proving the intersection of lattices is finitely generated over non-discrete valuation ring
I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of a subring of the hyperreals. It has a non-archimedean ...
- 131
4 votes
1 answer
138 views
Locally free coherent module over open disc
Let $(K,|.|)$ be a complete nonarchimedean field (Spherically complete if it is necessary). Let $D$ be the unit disc in the sense of Berkovich and $\mathcal{F}$ a locally free coherent sheaf. Can we ...
3 votes
1 answer
391 views
Schemes over formal disc with smooth generic fiber
Let $k$ be a field, and let ${\mathcal O}=k[[t]], {\mathcal K}=k((t))$, $D=Spec(\mathcal O)$, $D^*=Spec(\mathcal K)$. Let $X$ be a scheme of finite type over $D$ with smooth generic fiber (i.e. which ...
- 7,379
1 vote
1 answer
134 views
Localization of almost finite torsion $\mathcal{O}_C$-algebras
Let $C$ be the completion of an algebraic closure of $\mathbb{Q}_p$, and let $\mathcal{O}_C$ be its ring of integral elements. Let $A$ be an $\mathcal{O}_C$-algebra that is torsion and almost ...
- 953
0 votes
0 answers
104 views
Singular values and eigenvalues of matrices with coefficients in an ultrametric field
Consider a complete ultrametric field $(\Omega,|.|)$. We endow $M_n(\Omega)$ with the maximum norm. Given a matrix $G\in M_n(\Omega)$, recall that the singular values $\sigma_1\geq\dots \geq \sigma_n$ ...
3 votes
0 answers
236 views
What are non-archimedean norms on $\mathbb{R}$, whose restriction to $\mathbb{Q}$ is trivial?
I wonder if there is any classification result on non-archimedean norms on $\mathbb{R}$, with trivial restriction to $\mathbb{Q}$? Any references or examples would be welcomed! Some examples of such ...
- 281
7 votes
1 answer
374 views
Does a special property hold if the Archimedean property for reals doesn't hold?
Suppose $\mathbb{R}^e=A \cup B$ in which $A \cap B=\varnothing$ and there exist real numbers $a_0$ and $b_0$ such that $a_0 \in A$ and $b_0 \in B$. My question is, can we construct $a \in A$ and $b \...
6 votes
2 answers
479 views
Weak Archimedean property instead of Archimedean property
We say that a sequence $(z_n)$ of real numbers is a modulated Cauchy sequence, whenever there exists a function $\alpha:\mathbb{N} \rightarrow \mathbb{N}$ such that: $$ |z_i-z_j| \le \frac{1}{k} \quad ...
1 vote
0 answers
121 views
Cartan decomposition over a not-necessarily-discretely-valued field
Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and ...
- 73
1 vote
0 answers
198 views
Labelling non-Archimedean sets
I was reading papers 1 and 2 on numerosities. These present a way of comparing the size of sets as equivalences of the size of the set intersected with finite subsets. I am trying to extend the work ...
5 votes
0 answers
527 views
Stalks of nonarchimedean spaces as analytic rings
Let $(A,A^+)$ be an affinoid Tate ring, and let $x \in X=\operatorname{Spa}(A,A^+)$. When defining the stalks of the structure sheafs ${\mathcal O}_{X,x} = \varinjlim_{x \in U} {\mathcal O}_{X}(U) $ ...
- 151
4 votes
1 answer
223 views
Maximum modulus principle over the $p$-adic integers
Consider $\mathbb{Z}_p$ the $p$-adic integers. Let $f\in\mathbb{Z}_p[x]$ be an arbitrary polynomial in one variable. Write $f(x) = \sum_{k}a_kx^k$. Is it true that $\|f\|:= \max_k |a_k|_p = \sup_{t \...
- 1,607
3 votes
2 answers
330 views
Examples of non-splittable norms
Let $K$ be a complete non-archimedean field. A norm on a finite dimensional vector space $V$ is a function $| \cdot | : V \to \mathbf{R}$ which satisfies the usual norm properties (with the non-...
- 273
4 votes
0 answers
136 views
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 ...
- 1,068
2 votes
0 answers
211 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{\...
- 21
4 votes
1 answer
250 views
Partition of unity for analytic manifolds over non-Archimedean local fields
I am looking for a reference to the following fact which, I hope, is correct. Let $X$ be a compact analytic manifold over a non-Archimedean local field. Let $X=\cup_\alpha U_\alpha$ be a finite open ...
- 23k
5 votes
2 answers
617 views
Non-trivial extension of representations have same central character
Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
user515481
2 votes
0 answers
78 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$, ...
- 3,213
1 vote
1 answer
308 views
Formal series which are always zero
Let $(k, |\cdot|)$ be a complete field with a non-Archimedean norm, not necessarily algebraically closed. Define the Tate algebra as follows: \begin{align*} k \langle T_1, \dots, T_n \rangle = \{ \...
- 1,607
0 votes
0 answers
95 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 $...
- 1,607
1 vote
0 answers
140 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 ...
- 541
4 votes
0 answers
336 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}$ ...
- 303
2 votes
0 answers
230 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-...
- 61
0 votes
1 answer
151 views
Complete residue field of a point of type 5
Let $(F,|.|)$ be a complete algebraically closed field. Let $x$ be the point of type 5 corresponding to the unit open disc of the adic affine line over $F$. Can we obtain a concrete description of the ...
2 votes
0 answers
315 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}$. ...
- 21
0 votes
0 answers
123 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 ...
- 586
7 votes
0 answers
463 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 ...
- 4,555
5 votes
1 answer
465 views
A question on linear algebra over non-Archimedean local field
Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T_a\}_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$. After a choice of subsequence, is it ...
- 23k
6 votes
1 answer
448 views
Realization of the $p$-adic Steinberg representation as a subrepresentation
Let $G = \mathrm{GL}_n(F)$ where $F$ = non-archimedean local field. The Langlands Classification tells one that all irreducible admissible reps of $\mathrm{GL}_n(F)$ can be realized as (the unique ...
- 778
20 votes
1 answer
3k views
Mixing solids and liquids
Is there a nontrivial way to consider products of archimedean and non-archimedean spaces in the context of Clausen–Scholze's analytic geometry? Context: Last week during a conference in Essen (School ...
- 31.4k
4 votes
0 answers
171 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 $...
- 119
-5 votes
1 answer
254 views
Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones? [closed]
Below, we interpret divergent integrals as germs of partial integrals at infinity: $$\int_0^\infty f(x) dx=\operatorname{bigpart} \int_0^\omega f(x) dx$$ where $\operatorname{bigpart}$ means taking ...
- 10.4k
2 votes
1 answer
397 views
Are there "pathological convex sets" over ultravalued fields of char 2?
In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...
- 803
7 votes
0 answers
331 views
Analogs of the Weil conjectures for non-archimedian fields
Suppose that $X$ is a smooth and proper variety defined over a perfect non-archimedian valued field $k$ of characteristic $p$. Then one can consider the action of Frobenius on crystalline cohomology. ...
- 373
5 votes
1 answer
219 views
An example where the non-Archimedean tensor product of normed modules is only seminormed?
Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}_{\geq 0}$ which satisfies $$ | 0_R| = ...
- 153
2 votes
0 answers
315 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 ...
user478030
1 vote
1 answer
255 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 $k$-...
- 73
2 votes
0 answers
102 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 ...
- 71
1 vote
0 answers
138 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 $\...
- 10.4k
3 votes
0 answers
169 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 ...
- 10.4k
8 votes
2 answers
582 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 ...
1 vote
1 answer
212 views
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 ...
- 1,294
4 votes
0 answers
205 views
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 ...