Questions tagged [rigid-analytic-geometry]
rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields
243 questions
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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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2
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Perfectoid universal covers
It is often said, with varying degrees of rigor or enthusiasm, that every rigid space (say over $\mathbb{C}_p$) has a pro-etale cover which is 'topologically trivial' in some sense. For example, this ...
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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 ...
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Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?
The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
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$p$-adic Hodge Theory for rigid spaces, after P. Scholze
I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties.
This question is around the "Poincaré Lemma" in the paper.
Throughout, let $X$ be a proper smooth rigid ...
user113453
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 (...
27
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1
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Motivation for relative schemes: why should one work with schemes over a ringed topos?
Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
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26
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3
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A roadmap for understanding perfectoid spaces
Perfectoid spaces are this year's subject for the Arizona Winter School (link) and, as preparation, I am currently trying to understand the subject better. There are wonderful explanatory accounts (...
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2
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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 ...
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1
answer
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Why are there three kinds of non-archimedean geometry?
It may seem silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to ...
- 1,270
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3
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Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects)
The first one $\mathrm{Spec}\mathbb{C}[[t]]$ is a scheme, the second one $\mathrm{Spf}\mathbb{C}[[t]]$ is a formal scheme. In my mind they both realize an "infinite order infinitesimal neighbourhood ...
- 23.4k
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1
answer
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function field analogy and global/absolute geometry
The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...
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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
20
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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
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1
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$p$-adic Bott periodicity?
The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred ...
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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 ...
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0
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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 ...
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1
answer
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D-modules on rigid analytic spaces
Is there a good notion of holonomic $D$-modules on rigid analytic spaces?
- 491
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1
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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
14
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555
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Vanishing of rigid cohomology for affine varieties
Let $k$ be a perfect field of positive characteristic and denote by $K$ the field of fractions of the ring of Witt vectors over $k$.
Question: If $X$ is an affine variety over $k$, do the rigid ...
- 141
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2
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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 ...
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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
12
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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 '...
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1
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534
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Open immersion of affinoid adic spaces
If $R$ and $S$ are complete Huber rings with $\varphi: R \to S$ a continuous map, then is it true in general that if $\mathrm{Spa}(S, S^\circ) \to \mathrm{Spa}(R, R^\circ)$ is an open immersion of ...
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1
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Can a covering space of the $p$-adic disc split over the circle?
Let $D = {\rm Sp}\, \mathbb{C}_p\langle x\rangle$ be the affinoid unit disc over $\mathbb{C}_p$.
Is there an example of a connected finite etale cover of $D$ whose restriction to the "unit circle" ${\...
- 16.2k
12
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1
answer
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(pro)Étale cohomology of adic spaces and inverse limit
I am studying Peter Scholze's paper $p$-adic Hodge theory for rigid-analytic varieties and I am confused by the following.
Let $X$ be a finite type scheme over $\mathbb{C}_p$ (proper and smooth if ...
- 273
12
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1
answer
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Symplectic structures in rigid geometry
Let $K$ be a non-archimedean valued field (with any further adjectives attached as necessary). I'm looking for references or information about symplectic structures on rigid $K$-spaces.
For example, ...
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3
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Trivialisation of vector bundles on Stein spaces
Does every vector bundle on a Stein space have a finite local trivialisation?
Definitions:
Stein space means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of ...
- 1,450
11
votes
1
answer
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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 ...
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1
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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 ...
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11
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1
answer
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Closed image of curves under $p$-adic logarithm, Coleman integrals and Bogomolov
Disclaimer: my knowledge of $p$-adic analysis/geometry is minimal.
Consider a smooth, complete curve $C$ of genus $g$ over $\mathbb{C}_{p}$, denote by $J$ its Jacobian and consider the embedding $C\...
- 1,000
11
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3
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2k
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p-adic Poincaré Lemma
suppose $X$ is a proper and smooth rigid analytic variety over $\text{Spa}(k)$, with $k$ a non-archimedean field of characteristic zero.
One has the de Rham complex of analytic differential forms on $...
user113393
11
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0
answers
375
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Quasi-separated rigid-analytic space without a formal model?
Well, my question is slightly embarrassing. When learning rigid geometry (mostly from Bosch's book) I realized that I don't know the answer to the following basic question.
Question. Is there an ...
- 16.2k
11
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0
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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
10
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1
answer
2k
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Translation between formal geometry and rigid geometry
I'm reading a paper that translates between formal geometry and rigid geometry.
In particular, this paper begins with two rigid analytic spaces $A$ and $C$ (each coming from a scheme over $\mathbb{Z}...
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10
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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 ...
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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
10
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1
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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? ...
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10
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1
answer
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Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$
$\DeclareMathOperator\Spa{Spa}$Fargues's Theorem for $\Spa(C,O_C)$ states that the category of (mixed characteristic) shtukas with one paw at $x_C$ is equivalent to the category of Breuil-Kisin-...
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Zeros of $p$-adic power series and rationality
Let $K$ be a non-archimedean field with valuation ring $(V,\mathfrak{m})$, and $K\langle t_1,\ldots, t_n\rangle$ a Tate algebra of convergent power series.
Fix $f \in V\langle t_1,\ldots, t_n\rangle$....
user129156
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Detecting $k$-affinoid spaces by vanishing cohomology
The property of being an affine scheme can be tested against all quasi-coherent sheaves in the following sense: a noetherian scheme $X$ is affine iff $H^i(X,\mathcal{F}) = 0$ for all quasi-coherent $\...
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On the definition of the etale site of an adic space
I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces".
First ...
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Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?
Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum
$$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$
Now ...
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1
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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
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1
answer
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G-topological spaces and locales
Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...
user513306
9
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0
answers
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What lies between algebraic geometry and analytic geometry?
Algebraic geometry and analytic geometry are closely related (witness GAGA). But the latter still seems much "bigger" than the former. I'd like to be able to get from algebraic geometry to ...
- 64k
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Why do Coleman functions form a sheaf?
In section 4 of Ammon Besser's 2002 'Coleman Integration Using the Tannakian Formalism,' he defines abstract Coleman functions, which we can describe roughly as those functions which arise by iterated ...
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687
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Why are the open and closed adic discs defined the way that they are?
The closed adic disc is defined as $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$, and the open adic disc is defined to be the fiber $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
- 2,081
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327
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What role, if any, do Archimedean valuations play in adic spaces?
I've been reading about adic spaces, and I couldn't help but wonder what would happen to the theory if one included in the definition of $Spa$ Archimedean valuations as well...?
Is there a weird ...
- 2,081
9
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0
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268
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Kodaira embedding theorem for rigid analytic varieties
Kodaira embedding theorem can be regarded as a vast generalizaton of
the projectivity criterion for complex tori: indeed, the Riemann
conditions essentially say that the line bundle defined by the
...
- 4,070