Questions tagged [rigid-analytic-geometry]
rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields
243 questions
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Iwasawa logarithm and analytic continuation
I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$.
...
- 539
20
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3
answers
2k
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Étale homotopy type of non-archimedean analytic spaces
The following is likely all obvious to the experts. But since the field looks tricky to an outsider, maybe I may be excused for asking anyway.
I am wondering about basic facts of what would naturally ...
- 19.9k
10
votes
1
answer
504
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Picard group of Drinfeld upper half space
Let $K$ be a $p$-adic field and $\Omega^{(n)}_K$ the $n$-dimensional Drinfeld upper half space over $K$ (which is a rigid analytic space over $K$).
Is the Picard group of $\Omega^{(n)}_K$ known? ...
- 10.5k
8
votes
1
answer
726
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What are the automorphisms of a perfectoid Tate algebra?
Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a_1T+a_2T^2+\cdots$, where $a_1\in \mathcal{O}...
- 4,487
7
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1
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Are admissible open subsets of an affinoid space of countable type?
A rigid analytic space $Y$ over a complete non-archimedean valued field $k$ is said to be of countable type if it has a countable (possibly finite) admissible covering by affinoids over $k$.
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- 3,545
2
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1
answer
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Is X_0(p) a Mumford curve over $Q_{p^2}$
Let $p$ be a prime number and $X_0(p)/\mathbf{Q}$ be the classical modular curve for
$\Gamma_0(p)$. Let $\tilde{X}_0(p)/\mathbf{Z}$ be the projective arithmetic surface corresponding to the ...
- 7,609
17
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0
answers
953
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A functor of points approach to Berkovich analytic spaces
Is it possible to define a Berkovich analytic space via its functor of points?
Let $k$ be a complete non-Archimedean field, possibly the trivial one. I am tempted to define a Berkovich analytic space ...
- 171
7
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0
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673
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Etale cohomology of Berkovich spaces
Suppose $X/\mathbb{Q}$ is a reasonable smooth projective variety with interesting etale cohomology. For example, we can say $X$ is an elliptic curve. To what extent does it make sense to study the ...
- 71
2
votes
1
answer
164
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Terminology: Epimorphism in non-archimedean analysis
In their book "Non-Archimedean analysis", when BGR refer to an epimorphism in the context of $k$-Banach algebras do they actually require (or does it follow that) such maps are surjections? For ...
- 3,555
4
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1
answer
726
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How should we understand the relative interior in Berkovich spaces
I'm reading Berkovich's book on analytic spaces. The notion of relative interior confuses me. Is there anyway to see how it "looks like"? For instance, if $r <1$, what is the relative interior of
\...
- 125
4
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1
answer
469
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Spherical completions and flatness
Let $k$ be a non-Archimedean field. Does there exist a spherical completion $K$ of $k$ such that for any $k$-Banach space $X$, the natural map $X \to X \widehat{\otimes}K$ is an isometric embedding? ...
- 1,180
25
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2
answers
4k
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Rigid analytic spaces vs Berkovich spaces vs Formal schemes
I wonder if someone could explain briefly what is the relation between these 3 formal models, of a Berkovich space, a rigid analytic space and a formal scheme?
I have been working with formal schemes ...
- 251
40
votes
1
answer
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Why is Faltings' "almost purity theorem" a purity theorem?
My understanding of purity theorems is that they come in several flavors:
1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob_p$ are algebraic numbers all of whose ...
- 585
18
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1
answer
1k
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Why do rigid spaces have "not enough points"?
In Brian Conrad's notes
here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero ...
- 585
0
votes
0
answers
318
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morphism from adic spaces to schemes
Let $X:=Spa A$ be an affinoid adic space, and $\underline X $ the ringed space of $X$. Let $Y:=Spec B$ be an affine scheme, $f: \underline X \longrightarrow Y$ a morphism of ringed spaces.
How to ...
- 1,921
4
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1
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307
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scheme of generalizations
Hi,
I have the following problem. Let $\mathcal{O}$ be a valuation ring and $S=Spec(\mathcal{O})$, denote with $s$ the closed point and with $\eta$ the generic one. Let $X\rightarrow S$ be a proper, ...
- 43
12
votes
1
answer
885
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bornological vector spaces over a non-archimedean field
Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in '...
- 1,180
9
votes
1
answer
1k
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Reference Request: Vector bundles in rigid analytic geometry
In algebraic geometry it is well-known (see Hartshorne Exercise II.5.16 for example) that there is a 1-1 correspondence between rank $n$ (geometric) vector bundles $\pi\colon Y\to X$ on a scheme $X$ ...
- 3,545
13
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1
answer
2k
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Reference for rigid analytic GAGA
I'm looking for a reference for the following result.
Theorem. Let $K$ be a complete, non-archimedean field, and let $X/K$ be a projective scheme, with analytification $X^\mathrm{an}$. Then the ...
- 1,838
15
votes
1
answer
841
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How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$?
I apologize in advance if this question is terribly naive. I've just recently learned a bit of rigid analytic geometry with the hopes of understanding some basic facts about eigenvarieties.
In the ...
- 3,380
7
votes
0
answers
882
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Rigid Uniformization vs Grothendieck's Local Monodromy Theory
I've noticed that some interesting results about abelian varieties can each be proven using one of two ways: the theory of rigid uniformization of abelian varieties or Grothendieck's local monodromy ...
- 15.4k
10
votes
2
answers
876
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Uniqueness of analytic continuation in rigid analytic geometry
In classical complex analysis it is easy to prove that a meromorphic function has at most one analytic continuation (on an open connected subset of $\mathbb C$, say).
The problem of non-uniqueness of ...
- 121
2
votes
1
answer
348
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Modules with connection over $p$-adic laurent series rings
If $X$ is a smooth rigid analytic space over a $p$-adic field $K$ (of characteristic zero), then every coherent $\mathcal{O}_X$-module with integrable connection is locally free. In his paper "...
- 1,838
10
votes
1
answer
1k
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Analytic elements in non-archimedean geometry
Let $(k,|.|)$ be a complete non-archimedean valued field. Let $D$ be the open unit disc over $k$. (Anything I write could be adapted to the case of an open annulus.) The ring $\mathcal{O}(D)$ of ...
- 4,132
1
vote
0
answers
522
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Component group of Neron model of a parametrized abelian variety
Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an ...
- 15.4k
2
votes
0
answers
148
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Support of Tor over affinoid algebras
Suppose $k$ is a complete nonarchimedian field, $A$ is a $k$-affinoid algebra, and $M$ is a finitely presented $A$-module. Is the set
$\tau(M)= \left\{ x \in \mathrm{Sp}(A)\,\mathrm{with}\,\mathrm{...
- 13.1k
7
votes
1
answer
756
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$p$-adic uniformization not from the Drinfel'd spaces?
It seems that when we talk about the $p$-adic uniformization, we typically mean those uniformized by either the Drinfel'd upper spaces (for which we think of the examples of Mumford curves and some ...
- 1,305
5
votes
1
answer
790
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Base Change for Eigenvarieties
Let $E/F$ be a Galois extension of number fields, and $G$ a reductive group over $F$. If Langlands Base Change is known for $G/F$ and $G/E$, and moreover the eigenvarieties for $G/F$ and $G/E$ have ...
- 4,790
2
votes
0
answers
455
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reference for p-adic Stein spaces
Hi,
I'm looking for a reference in english for p-adic Stein spaces. The usual referneces I come across are all in german.
Thanks
- 2,842
16
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1
answer
1k
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D-modules on rigid analytic spaces
Is there a good notion of holonomic $D$-modules on rigid analytic spaces?
- 491
3
votes
1
answer
495
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universal finite differential module of affinoid algebra
Let $k$ be a value field (archimedean), for example $k = \mathbb{Q}_p$, the p-adic field.
The free Tate algebra is $$ T_n := \left\{ \ \sum a_I X^I, \ a_I \in k, \ a_I \rightarrow 0 \text{ as } |I| \...
- 1,109
11
votes
0
answers
454
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Is the Gouvea-Mazur problem related to symmetric square $L$-functions?
Here's an idea that I've found appealing but have never been able to get anywhere with.
One way to frame the Gouvea-Mazur question (for lack of a better term, since the original conjecture by the ...
- 2,783
11
votes
1
answer
815
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Consequences of the geometric properties of the eigencurve
The eigencurve $\mathcal{E}$ is a rigid-analytic space parametrizing certain $p$-adic families of modular forms and associated Galois representations. By constructing an auxiliary reduced rigid curve ...
- 163
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0
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What is the nature of the zero locus of a section of a coherent sheaf?
Suppose that $X$ is a reduced rigid space and $\scr{F}$ is a coherent sheaf on $X$. For a section $f\in {\scr F}(X)$, the zero locus of $f$ is the set of points $x\in X$ at which $f$ vanishes in the ...
- 2,783
6
votes
1
answer
737
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Tate models for semistable algebraic varieties with mixed reduction over a local field
It's known that if $A$ is an abelian variety of totally multiplicative reduction over a p-adic field K, then, after taking a finite field extension, it becomes isomorphic, as a rigid analytic group, ...
- 7,972
3
votes
0
answers
484
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Sheaf of power-bounded elements in rigid analytic geometry
Let $k$ be a field with a non-archimedean complete valuation $|\ |$, $X$ a reduced rigid analytic space over $k$. The presheaf $\mathcal{O}^0$ which to an affinoid $U$ of $X$ attaches the ring $\...
- 26.1k
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0
answers
2k
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Generalized GAGA
So, I have heard GAGA works for Rigid Analytic spaces. I know next to nothing about this, but it made me curious as to whether there are any other contexts in which GAGA "works". Of course, this is a ...
- 373
20
votes
1
answer
2k
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Are flat morphisms of analytic spaces open?
Let $f:X\to Y$ be a morphism of complex analytic spaces. Assume $f$ is flat (or, more generally, that there is a coherent sheaf on $X$ with support $X$ which is $f$-flat). Is $f$ an open map?
The ...
- 19.6k
32
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1
answer
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Structure on $X(k)$ for separated finite type alg. space $X$, for complete valued $k$.
Let $k$ be a field complete with respect to a non-archimedean absolute value, and $X$ a separated algebraic space of finite type over $k$.
If $X$ is a scheme then $X(k)$ inherits a natural (...
11
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1
answer
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Do Berkovich homogenous spaces exist?
Let G be a k-analytic group, and let H be a closed subgroup of G. Then does there exist a k-analytic space, which can be reasonably called the quotient G/H?
Commentary: I realise that I am not being ...
- 8,866
1
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1
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762
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What are in units of an affinoid algebra?
Suppose that $K$ is a complete local field and $A$ is an affinoid $K$-algebra. Is there a known way to produce an explicit description of the units of $A$?
Here is what I already know: write $A^\circ$...
- 3,545
13
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2
answers
2k
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Cohomology of rigid-analytic spaces
Let $R$ be a complete discrete valuation ring and let $K$ be its field of fractions. Suppose $X$ is a smooth rigid-anaytic space over $K$. Often it is convenient to have a model of $X$ whose ...
- 4,487
8
votes
3
answers
527
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Weierstrass points on rigid-analytic surfaces
Does a rigid-analytic surface defined over a nonarchimedean complete field have Weierstrass points (if its genus is big enough let's say)? Is there a good reference that (ideally) lists theorems for ...
- 83