All Questions
Tagged with rigid-analytic-geometry nt.number-theory
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
35
votes
0
answers
1k
views
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 $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
- 64k
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
32
votes
1
answer
2k
views
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 (...
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
12
votes
1
answer
535
views
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 ...
- 503
11
votes
1
answer
815
views
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
10
votes
1
answer
2k
views
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}...
- 949
9
votes
0
answers
687
views
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
9
votes
0
answers
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 ...
- 2,081
8
votes
1
answer
726
views
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
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
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 ...
- 37k
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 ...
- 15.4k
6
votes
1
answer
737
views
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
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 ...
- 61
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 ...
- 323
5
votes
1
answer
966
views
simple questions on topological rings arising in the context of Perfectoid Spaces
(I apologize in advance for these simple questions, I am a beginner trying to go through Scholze's paper Perfectoid Spaces).
Let $(R, R^+)$ be an affinoid $k$-algebra as defined in Scholze's paper ...
- 785
5
votes
1
answer
790
views
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
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 ...
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 $...
- 51
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 :...
- 37k
4
votes
1
answer
1k
views
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
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
0
answers
422
views
Cyclotomic Extension of a Perfectoid Space
Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more ...
- 41
3
votes
1
answer
180
views
Approximating $p$-adic power series by polynomials
Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...
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
200
views
Kernel of a map of Tate algebras
Let $A$ and $B$ be a pair of noetherian Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow ...
- 2,907
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
281
views
The closed unit adic disk
I am reading the Scholze-Weinstein Berkeley lecture notes on "Perfectoid Spaces", and in particular I am stuck trying to understand the closed adic unit disk, which is the second example of ...
- 2,907
3
votes
0
answers
331
views
Etale cohomology of rigidification
Let $K/\mathbb Q_p$ be a discretely valued non-archemedean field, let $X$ be a smooth scheme over $\mathcal O_K$. To $X$ one can associate two rigid-analytic spaces over $K$:
1) the analytification $...
- 790
2
votes
1
answer
381
views
Reduced complete Tate ring which is not uniform?
Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...
- 217
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
1
vote
0
answers
348
views
rigid analytic geometry positive characteristic
I am a beginning graduate student. I have the following basic question I am very confused about:
Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
- 147
1
vote
0
answers
522
views
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
0
votes
0
answers
64
views
Weierstrass subdomain of $\DeclareMathOperator\Spm{Spm}\Spm \mathbb{Q}_p$
I am trying to understand Weierstrass subdomains of $\Spm\DeclareMathOperator\QP{\mathbb{Q}_p}\QP$.
Recall that a Weierstrass algebra of an affinoid space $\Spm A$, where $A$ is a Banach algebra with ...
- 2,907