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Questions tagged [rigid-analytic-geometry]

rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields

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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 $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
Tim Campion's user avatar
17 votes
0 answers
953 views

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 ...
Martin Ulirsch's user avatar
14 votes
0 answers
555 views

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 ...
Niels's user avatar
  • 141
12 votes
1 answer
579 views

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, ...
Mr. Chip's user avatar
  • 179
11 votes
0 answers
375 views

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 ...
Piotr Achinger's user avatar
11 votes
0 answers
454 views

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 ...
Ramsey's user avatar
  • 2,783
10 votes
0 answers
269 views

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 $f \in V\langle t_1,\ldots, t_n\rangle$....
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10 votes
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409 views

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 $\...
msteve's user avatar
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9 votes
0 answers
577 views

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 ...
Tim Campion's user avatar
9 votes
0 answers
391 views

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 ...
pupshaw's user avatar
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9 votes
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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 $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
Andrew NC's user avatar
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9 votes
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327 views

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 ...
Andrew NC's user avatar
  • 2,081
9 votes
0 answers
268 views

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 ...
Dima Sustretov's user avatar
9 votes
0 answers
1k views

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 ...
Ramsey's user avatar
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8 votes
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Looking for basic example in the theory of adic spaces

I have heard that there exist examples of Huber pairs $(A,A^{\circ})$ and $(B,B^+)$ such that $\operatorname{Spa}(B,B^+)$ is a rational open of $\operatorname{Spa}(A,A^{\circ})$ and such that $B^+ \...
sdr's user avatar
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8 votes
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Foundational Questions on Adic Spaces

There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...
gdb's user avatar
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8 votes
0 answers
354 views

Serre's examples in rigid analytic geometry

Over $\mathbb{C}$, we have the following phenomenon: there exist algebraic varieties whose etale homotopy types are isomorphic but the homotopy types of their analytifications are not. Such examples ...
user avatar
8 votes
0 answers
310 views

Disconnectedness in Rigid Analytic Geometry: a technical question

I have a technical question in rigid analytic geometry, in the sense of Tate. Let $B$ be an affinoid space over $\mathbb Q_p$, $Z$ a rigid analytic variety with a flat surjective map $f:Z \...
Joël's user avatar
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8 votes
0 answers
518 views

$p$-adic uniformisation of abelian varieties

In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement: Let $A$ over $\mathbf{Q}_p$ be an abelian variety ...
cannonball's user avatar
7 votes
0 answers
203 views

Finite generation and finite presentation over a truncated valuation ring: is there an easier proof?

Let $K^+$ be a valuation ring which is $\pi$-adically complete for some pseudouniformizer $\pi$. Nagata 053E proved that every finitely generated and flat (equivalently, torsion-free) $K^+$-algebra is ...
Piotr Achinger's user avatar
7 votes
0 answers
416 views

$F$-isocrystals defined via a lift of a scheme

Let $X$ be a smooth affine scheme over a finite field $k$. Then there exists a smooth affine formal scheme $\mathfrak{X}$ over $W(k)$ with a lift $\sigma$ of the Frobenius. A convergent $F$-isocrystal ...
curious math guy's user avatar
7 votes
0 answers
257 views

Etale maps from smooth affinoids to balls

Let $X$ be a smooth affinoid rigid space over a nonarchimedean field $K$. Does $X$ admit an etale map to a $K$-affinoid ball? The answer is almost certainly "no" in general, but I don't know ...
David Hansen's user avatar
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7 votes
0 answers
574 views

Rigid analytic varieties vs rigid spaces

In rigid analytic geometry, some sources refer to "rigid spaces", where others refer to "rigid analytic varieties". Do these two terms stand for the same thing, or is there a ...
Alexander Betts's user avatar
7 votes
0 answers
235 views

Rigid cohomology with support and dagger spaces

Let $K$ be a $p$-adic field with residue field $k$, and $X$ a variety over $k$. The rigid cohomology of $X$ over $k$ can be described very neatly using Grosse--Kloenne's notion of dagger spaces: embed ...
David Loeffler's user avatar
7 votes
0 answers
673 views

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 ...
none's user avatar
  • 71
7 votes
0 answers
882 views

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 ...
David Corwin's user avatar
  • 15.4k
7 votes
0 answers
2k views

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 ...
Ian M.'s user avatar
  • 373
6 votes
0 answers
155 views

Resolution property in rigid analytic geometry

I am not a rigid analytic geometrer, so I apologise if the question is trivial, but I can't find an answer anywhere myself. I'm trying to understand in what ways (rigid) analytic geometry compares to ...
Tim's user avatar
  • 1,131
6 votes
0 answers
356 views

Coherent cohomology of the generic fiber of Lubin-Tate space vs. of Lubin-Tate space considered rationally?

I am trying to compare the coherent cohomology of the generic fiber of Lubin-Tate space to the coherent cohomology of Lubin-Tate space considered rationally, and I am going in circles! I would be very ...
Catherine Ray's user avatar
6 votes
0 answers
290 views

Is being perfectoid a local property?

In his 2012 CDM proceedings, Peter Scholze mentions the following open question: Let $K$ be a perfectoid field and $(A,A^+)$ a complete affinoid $K$-algebra. Suppose there exists a cover of $X = \...
Kim's user avatar
  • 4,164
6 votes
0 answers
232 views

Complete characteristic p perfect Tate rings are uniform?

In Lemma 7.1.6 of his lecture notes on perfectoid spaces, Bhatt states that every complete characteristic p perfect Tate ring $A$ is uniform. In the proof he uses the Banach open mapping theorem on ...
Corvin Paul's user avatar
6 votes
0 answers
421 views

What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
Will Dukeminier's user avatar
5 votes
0 answers
438 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) $ ...
Theodor's user avatar
  • 151
5 votes
0 answers
197 views

Bezout-type theorem for $p$-adic analytic plane curves

Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
Ashvin Swaminathan's user avatar
5 votes
0 answers
132 views

Is $\mathbf{C}_p(X)$ self-dual?

Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
Luiz Felipe Garcia's user avatar
5 votes
0 answers
556 views

Theorem 7.11 in Scholze's $p$-adic Hodge Theory

I was trying to understand the statement and proof of Theorem 7.11 in Scholze's paper "$p$-adic Hodge Theory for Rigid-Analytic Varieties". I'll reproduce part of the statement below: Let $...
Kush Singhal's user avatar
5 votes
0 answers
122 views

Are affinoid algebras over nontrivially valued fields Jacobson?

It is well-known that for any field $k$ with valuation the Tate algebra $k\{T_1,\dots,T_n\}$ is Jacobson (see Bosch-Güntzer-Remmert for nontrivial valuations; for trivial valuations those are just ...
Wojowu's user avatar
  • 28.2k
5 votes
0 answers
405 views

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 :...
David Loeffler's user avatar
5 votes
0 answers
180 views

Homotopy of rigid analytic spaces

Let $K$ be a complete non-Archimedean valued field (I think the valuation does not have to be discrete). For a paracompact strictly $K$-analytic space, I have seen at least two definitions of ...
user avatar
5 votes
0 answers
1k views

Formal GAGA and étale cohomology

Let $\mathfrak{X}$ be a $p$-adic flat formal scheme over $\mathbf{Z}_p$, whose special fiber has an ample line bundle. Then $\mathfrak{X}$ is algebraizable, that is there exists an algebraic $\mathbf{...
user avatar
5 votes
0 answers
188 views

Ring of functions of generic fiber of affine special formal schemes

Fix $R$ a complete DVR. Recall from Berkovich's Vanishing Cycles for Formal Schemes II paper that we have a class of special formal schemes which are not topologically of finite type over $\...
Joe Berner's user avatar
5 votes
0 answers
1k views

Compatibility of formal completion and rigid analytic generic fiber

Let $R$ be a complete valuation ring of rank $1$ (e.g., a complete discrete valuation ring) and let $K$ be its field of fractions. Consider a proper $R$-scheme $X$ that is, say, normal (if needed). ...
O-Ren Ishii's user avatar
4 votes
0 answers
117 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 ...
Jef's user avatar
  • 984
4 votes
0 answers
205 views

Notion of connected components for $\mathbb{Q}_p$-points of algebraic variety

Is there an interesting notion of connected components for the $\mathbb{Q}_p$-points of an algebraic variety over $\mathbb{Q}_p$? By "interesting" I mean a notion satisfying the following. ...
Jacques's user avatar
  • 563
4 votes
0 answers
321 views

Can we see the completion of a scheme along a subscheme as an adic space?

Classically, formal schemes were invented to study completions of schemes along closed subschemes. Eventually, people started using them for more arithmetical reasons. (I.e., to study non-archimedean ...
Gabriel's user avatar
  • 773
4 votes
0 answers
149 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 $...
Takagi Benseki's user avatar
4 votes
0 answers
364 views

(Co)limits of adic spaces

Which (co)limits exist in the category of adic spaces ? Also, can we impose adjectives such as "noetherian" or "quasi-compact", etc., to get more (co)limits ? I know that finite ...
Dat Minh Ha's user avatar
  • 1,516
4 votes
0 answers
843 views

An attempt to define partial properness and compactification for some maps between analytic spaces

The paper Étale cohomology of diamonds defines partial properness and compactification for maps between v-sheaves, and in particular for perfectoid spaces and rigid-analytic spaces. Recently when ...
Longke Tang 唐珑珂's user avatar
4 votes
0 answers
281 views

nearby cycles map for affine formal schemes

Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\...
ali's user avatar
  • 1,093
4 votes
0 answers
130 views

Existence of a "p-adic Mahler measure" or alternatively, the converge of a p-adic sequence

Let $f \in \mathbb Z_p[[t]]^\times$ be an invertible power series and let $\log_p$ be the p-adic logarithm with the normalization that $\log p = 0$. Consider the sequence: $$a_n = \frac{1}{p^{n-1}}\...
Asvin's user avatar
  • 7,746