31
votes
Accepted
Are rigid-analytic spaces obsolete, since adic spaces exist?
There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications?
For the first question, it'...
- 37k
31
votes
Are rigid-analytic spaces obsolete, since adic spaces exist?
My opinion (knowing that this is somehow a matter of taste) is that the rigid-analytic viewpoint is more or less obsolete, and that one should better use Berkovich, Huber or Raynaud instead (each of ...
- 1,584
21
votes
Accepted
Berkovich space including both archimedean and non-archimedean worlds
The definition of analytic space over $\mathbf{Z}$ was given by Berkovich in his foundational book "Spectral theory and analytic geometry over non-Archimedean fields" (see the beginning of section 1.4 ...
- 4,132
12
votes
Accepted
Hahn’s theorem on ordered fields
This theorem, which extends Hahn's embedding theorem for ordered abelian groups to ordered fields, has a complicated history that makes it difficult to attribute it to any single author. However, by ...
- 6,453
12
votes
Mixing solids and liquids
Good question!
I think the real context for the question was whether certain objects that are implicit in work of Darmon (and collaborators) could exist within this framework of analytic geometry. ...
- 21.3k
9
votes
Accepted
Why does $\mathbb C_p$ not contain the periods?
Consider the Tate motive $\mathbb Q(1)$. Its de Rham realization is simply $\mathbb Q$ (with the filtration $F^{-1}\mathbb Q=\mathbb Q$ and $F^{0}=0$) and its Betti realization is $2\pi i\mathbb Q$. ...
- 10.9k
8
votes
Accepted
Examples of non-splittable norms
All norms on finite-dimensional spaces over $K$ are splittable if, and only if, the field $K$ is maximally complete. You can find this result (and much more) in the paper by Boucksom–Eriksson “Spaces ...
- 4,132
8
votes
Accepted
Maximum modulus principle over the $p$-adic integers
No. Let $K$ be a nonarchimedean discrete valued field with a finite residue of order $q$. Let $f(x) = x^q - x$ and $\pi$ be a uniformizer in $K$ (generator of $\mathfrak m_K$). Then $|\!|f|\!| := \...
- 50.6k
7
votes
Accepted
Literature on non-Archimedean analogues of basic complex analysis results
Benedetto has a textbook that discusses basic $p$-adic analysis, although his aim is to study $p$-adic dynamics. And it's for a single variable. But might be a good place to get some information.
...
- 47.4k
6
votes
Accepted
Fontaine, J.-M.; Illusie, L. p-adic periods------Does any one have the following article?
It seems clear that there is no online version available, but a library can probably get a PDF copy through interlibrary loan if that's an option for you. The detailed listing is here. Note too ...
- 52.9k
6
votes
Formally real fields with unique non-Archimedean ordering
Yes, such fields exist. Let $T$ be the first-order theory in the language of fields with an extra constant $c$, axiomatized by
the axioms of fields,
every element or its negative is a sum of 4 ...
- 47.1k
6
votes
Accepted
How does an analytic space correspond to a $p$-adic Banach space
Without reading much beyond the pages adjacent to page 6, but looking at [ST3] (especially seeing the references [BGR] and [NFA]), I believe Berger-Colmez are using $K_n$-analytic spaces in the sense ...
- 505
6
votes
Accepted
A question on linear algebra over non-Archimedean local field
$\def\FF{\mathbb{F}}\def\GL{\text{GL}}$This is basically the same thing YCor sketches in his comment: Let $R$ be the ring of integers of $\FF$. Let $K$ be the group of invertible matrices with entries ...
- 156k
6
votes
Accepted
Formal series which are always zero
Yes (assuming $k$ infinite: $k$ finite yields easy counterexamples), by a simple argument of elementary analysis.
Case $k$ infinite, discrete: then $f$ is a polynomial vanishing on $k^n$, hence is ...
- 63.9k
6
votes
Non-trivial extension of representations have same central character
You probably should assume all representations are smooth.
You hope to show if $\pi_1$ and $\pi_2$ have two different central characters $\phi_1$ and $\phi_2$ then any extension $0\to \pi_1\to \pi\to \...
- 3,054
6
votes
Partition of unity for analytic manifolds over non-Archimedean local fields
It follows from Lemma 1 (part (2)) on page 7 in http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bernst_Lecture_p-adic_repr.pdf
In the non-compact case, I think that you need to ...
- 2,639
5
votes
Accepted
Is $\mathbb{A}_k^n(k)$ dense in the Berkovich analytification of $\mathbb{A}_k^n$?
If $k$ is not algebraically closed, then $\mathbb A^n_k(k)$ is not doing to be dense with $\mathbb A_k^n$. I'll show this for $n=1$ for simplicity. Take any point $P$ in $\mathbb A^1_k$ with a residue ...
- 28.2k
5
votes
Accepted
Image of a finite dimensional complex representations of $GL_n(\mathcal{O})$
Call the representation $\pi$. Let $U$ be a neighbourhood of the identity in $\operatorname{GL}(V)$ that contains no non-trivial subgroup. Then the pre-image of $U$ under $\pi$ is a neighbourhood of ...
- 12.9k
5
votes
Accepted
Complete residue field of a point of type 5
A point of type 5 corresponds to a valuation of rank 2, so I am not sure if the meaning of completion is completely clear here. I think that the usual thing is to complete with respect to the ...
- 4,132
5
votes
Accepted
Non-trivial extension of representations have same central character
@KentaSuzuki's answer is probably the best way of thinking about it, but here's another argument in line with what you wanted.
Permit me to write $\phi_1$ rather than $\phi$ for the central character ...
- 12.9k
4
votes
Berkovich space including both archimedean and non-archimedean worlds
You should take a look at the paper by Jérôme Poineau:
La droite de Berkovich sur Z, Astérisque n° 334 (2010)
and other papers by the same autor, like
Espaces de Berkovich sur Z : étude locale, ...
- 1,386
4
votes
Accepted
Tropical charts (coordinates) and differential forms in non-archimedean geometry
I will try to explain what it is a tropical chart on an algebraic variety over a non-archimedean field $K$ (complete with respect to a non-archimedean absolute value, algebraically closed by ...
- 1,386
4
votes
Accepted
Significance of integrally closed in an affinoid algebra
The adic spectrum of $(R,R^+)$ is a set of continuous valuations on $R$ having norm $\leq 1$ on $R^+$. It is clear from the definition and basic properties of integral closures that relaxing the ...
- 56
4
votes
Fontaine, J.-M.; Illusie, L. p-adic periods------Does any one have the following article?
The desired paper is freely available on Fontaine's webpage! Here is the first part, and here is the second.
- 1,742
3
votes
Accepted
Raynaud's universal Tate elliptic curves
Q1: Michel Raynaud, Géométrie analytique rigide d’après Tate, Kiehl..., Bull. Soc. Math. France, Mémoire 39-40, p. 319-327 (1974).
- 188k
3
votes
Accepted
Definition of model functions and their density in $C^0(X^\text{an})$
As alluded to in the question, the space $\mathcal{D}(X)_{\mathbb{Z}}$ is indeed the space of model functions arising from integral divisors.
Now, in order to deduce that $\mathcal{D}(X)_{\mathbb{Q}}$ ...
- 582
3
votes
Accepted
finite number of vertices of the polyhedron of variation of an invertible function on a Berkovich curve
It is not true in general that $P$ is finite. To see this, take an open disk $D$ and a non-zero function $f$ on it with infinitely many zeroes. In this case, your polyhedron of variation is infinite: ...
- 4,132
3
votes
Reference to basic facts on non-Archimedean local fields
These are sufficiently elementary that they're not likely to have a proof that is citable.
YCor has answered 1 in the comments but I'll repeat it here for completeness.
Proof of 1: As $M$ is open it ...
- 8,866
3
votes
Accepted
Theory of extensions of non-archimedian local fields
See Fesenko and Vostokov - Local fields and their extensions. (i) is Proposition 3.3(2). (ii) is Proposition 3.2(1). (iii)(1) is Proposition 3.5(1) (and, yes, $b$ may be chosen as a uniformiser). ...
- 12.9k
3
votes
Accepted
perfectoid field of characteristic $p$
Write $I$ for the image of $\operatorname{Tr}_{L'/L} \colon \mathfrak m' \to \mathfrak m$, which is an ideal because $\operatorname{Tr}_{L'/L}$ is $\mathcal O_L$-linear.
(1) The first sentence is ...
- 28.7k
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