All Questions
Tagged with rigid-analytic-geometry arithmetic-geometry
36 questions
41
votes
2
answers
3k
views
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 ...
- 843
40
votes
1
answer
14k
views
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
32
votes
1
answer
8k
views
$p$-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
23
votes
1
answer
1k
views
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 ...
- 19.9k
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 ...
- 141
11
votes
1
answer
406
views
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
10
votes
1
answer
564
views
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-...
- 435
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 $f \in V\langle t_1,\ldots, t_n\rangle$....
user129156
8
votes
1
answer
339
views
On actions of finite groups on adic spaces
Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
- 541
7
votes
1
answer
1k
views
What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?
Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Actually I do not know many details about it. It says an analytic stack is a sheaf $\...
- 83
7
votes
1
answer
480
views
Rigid versus log-rigid cohomology for semistable varieties
If $K$ is a p-adic field, with maximal unramified subfield $K_0$, and $X$ is a proper semi-stable $O_K$-scheme, then there's a canonical way to make the special fibre $X_k$ into a log-scheme; and ...
- 37k
7
votes
1
answer
756
views
$p$-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
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 ...
- 3,539
5
votes
1
answer
362
views
On the noetherianess of some subalgebras of an affinoid algebra
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
- 541
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{...
user92332
4
votes
1
answer
545
views
On the local properties of rigid analytic varieties
Let $K$ be a non-archimedean field complete with respect to a discrete valuation with ring of integers $\mathcal{R}$, uniformizer $\pi$ and residue field $k$. Consider an affinoid analytic $K$-variety ...
- 541
4
votes
1
answer
369
views
On a consequence of the Gerritzen-Grauert Theorem
Let $K$ be a local field of characteristic zero and $X$ an affinoid rigid space over $K$. Let $U\subset X$ be an affinoid subdomain, and consider a finite family of points $\{p_{1},\cdots, p_{n}\}\...
- 541
4
votes
1
answer
393
views
Topological and algebraic covering spaces in Berkovich geometry
Let $k$ be a complete, non-archimedean field, and $X$ a Berkovich space over $k$ (as nice as you like, for arguments sake let's say strictly $k$-analytic, good, and geometrically connected). As ...
- 1,838
4
votes
1
answer
184
views
Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operators" by Coleman and Iovita
Let $G$ be an abelian variety with good reduction over a finite extension $K$ of $\mathbb{Q}_p$. In equation (2.4) on page 179 of my edition of "The Frobenius and monodromy operators for curves ...
- 658
4
votes
1
answer
259
views
On inverse limits of $\pi$-adically complete algebras
Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}_{p}$. Let $\{ A_{n}\}_{n\in\...
- 541
4
votes
1
answer
307
views
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
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}}\...
- 7,746
3
votes
1
answer
187
views
Reference Request: Preservation of étale maps under rigid analytic GAGA
Let $K$ be a finite extension of $\mathbb{Q}_p$. As the title says, I am looking for a reference in which it is shown that given an étale map $f:X\rightarrow Y$ between smooth algebraic $K$-varieties, ...
- 541
3
votes
1
answer
469
views
Adic generic fiber of a small formal scheme in the sense of Faltings
$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic ...
- 31
3
votes
1
answer
275
views
complement of "good reduction" points in p-adic shimura varieties
assume that $X$ is Siegel Shimura variety defined over $\mathbb{Z}_p$, you can take its p-adic formal completion $\mathfrak{X}$,and than take it's adic generic fiber $\mathcal{X}$ and get an adic ...
- 1,093
3
votes
1
answer
474
views
Crystalline comparison for rigid-analytic varieties
Let $k$ be a finite extension of $\mathbb{Q}_p$. In this paper, Scholze proves an analogue of de Rham comparison for proper smooth rigid-analytic varieties over $k$. He also says:
...it should be ...
user138661
3
votes
1
answer
385
views
Overconvergent modular forms and the level at $p$
I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.
The ...
- 241
3
votes
1
answer
171
views
On the stability of having a normal formal model under finite extensions of the base field
Let $K$ be a finite extension of the $p$-adic numbers with valuation ring $\mathcal{R}$ and uniformizer $\pi$. Consider a smooth and connected rigid $K$-variety $X=Sp(A)$ and assume that the affine ...
- 541
3
votes
1
answer
147
views
Bounded torsion of quotients of affine formal models
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
- 541
3
votes
1
answer
263
views
On the exactness of some completed tensor products
Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/...
- 541
3
votes
1
answer
312
views
Geometric line bundles on the Tate curve
Let $E_q$ be the rigid analytic Tate elliptic curve over a complete algebraically closed non-archimedean field $K$ of mixed characteristic $(0,p)$, with parameter $q\in K^{\times}$ with $|q|<1$.
...
user178246
3
votes
0
answers
82
views
When is a coherent sheaf on an algebraizable space algebraizable?
Let $\mathcal{X}$ denote an algebraizable rigid analytic space over $\text{Spm}(\mathbb{Q}_p)$,
i.e. there exists a finite type $\mathbb{Q}_p$-scheme $X$ such that $\mathcal{X}$ is its rigid ...
- 2,907
2
votes
0
answers
110
views
Galois action on the cohomology of a curve over a local field with bad reduction
Let $C/\mathbb Q_p$ (or a p-adic local field more generally) be a smooth projective curve with split semistable reduction over $\mathbb Z_p$. What can we say about the action of the Galois group $\...
- 7,746
2
votes
0
answers
166
views
Theorem on formal functions when the initial data is a proper map of formal schemes
Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$.
Set $S_0=\{x\}$ be a closed point of $S$ and $...
- 1,066
2
votes
0
answers
257
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-...
user141691
1
vote
0
answers
177
views
L-function in p-adic spaces
I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate ...
- 4,428