27
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 ...
- 1,454
27
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'...
- 31.5k
20
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 ...
- 3,697
11
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 ...
- 5,299
10
votes
Is there a notion of pure dimension for Berkovich analytic space?
Yes, there is a good notion of dimension, due to Berkovich and developed in my article, as mentioned in the two answers above.
Concerning your question about GAGA principle for pure dimension, the ...
- 1,454
9
votes
Berthelot functor, rigid analytic space
The setup of the question is not general enough:
(i) you mean to work with Spf rather than Spec,
(ii) Raynaud's construction doesn't apply to the formal scheme Spf($A$) for such $A$,
(iii) the ...
8
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 $2\pi i\mathbb Q$. ...
- 9,510
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.
...
- 42.3k
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 ...
- 50.8k
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 ...
- 39.3k
6
votes
Accepted
Is there an exponential map on (Hahn) ordered fields?
There is no such exponential map. This was demonstrated in:
F.-V. Kuhlmann, S. Kuhlmann, S. Shelah, Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997) 3177–3183.
- 5,299
5
votes
A question on non-archimedian Fourier transform
Can one try to show the negative answer for n=2 as follows. The question is equivalent to existence of a nonzero distribution σ on a 4 dimensional space $K^4$ such that both both σ and FT(σ) are ...
- 1,253
5
votes
Accepted
A question on non-archimedian Fourier transform
Actually, the answer is indeed negative and the explanation is very simple. Namely, for $n=2$ let $\phi$ be the delta-function of the space of matrices whose second row is zero (considered as a ...
- 6,677
5
votes
Accepted
Rigid analytic geometry in characterstic 0 vs positive characteristic
Resolution of singularities for rigid analytic varieties of equal characteristic zero follows from resolution of singularities for schemes of characteristic zero (Nicaise, A trace formula for rigid ...
- 1,380
5
votes
Non-Archimedean non-standard models for R
Take for $S$ the field $F(t)$ with $F$ being the algebraic closure of $\mathbb Q$ inside $\mathbb R$. Equip it with the unique ordering for which $t-x>0$ for every integer $x$, and take a maximal ...
- 1,454
5
votes
Accepted
Relations between two definitions of non-archimedean analytic spaces
Let me give more a few more details than in nfdc23's comment. The most general definition of a analytic space is the one that you find in Berkovich's IHES paper. One requirement is that for every ...
- 3,697
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 ...
- 7,931
5
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 ...
- 495
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 ...
- 23.7k
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 ...
- 1,321
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.
- 1,637
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, ...
- 1,321
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
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: ...
- 3,697
3
votes
Accepted
Norm vs A-norm in non-Archimedean Functional Analysis
I guess that putting an archimedean norm on a vector space over a nonarchimedean field gives just an uncorrelated product of something archimedean with something nonarchimedean. Number theorists ...
- 1,380
3
votes
Accepted
the structure on the value group sort of a C-minimal field
I think the answer is no. Consider an algebraically closed valued field in the three sorted language. Using a relative quantifier elimination argument, any o-minimal expansion of the value group ...
- 175
3
votes
A question on non-archimedian Fourier transform
Sorry but I would like to change my vote and would now argue for a positive answer.
I am afraid I made a mistake when saying that I know a distribution on $sl(2)$ supported on the nilcone whose FT ...
- 1,253
3
votes
Accepted
is every point of a Berkovich space a Shilov point?
No. Saying that the point $x$ lies in the Shilov boundary of $\mathcal{M}(A)$ gives strong restrictions on the completed residue field $\mathscr{H}(x)$. In your case, its residue field $\widetilde{\...
- 3,697
3
votes
Accepted
The target of a regular function in Non-archimedean analytic geometry
If you want to think of $f$ in a way not too far from complex intuition, you should rather consider the induced morphism $\varphi$ from $X$ to the Berkovich affine line. If $x\in X$ then $\varphi(x)$ ...
- 1,454
3
votes
Polynomial inequalities in ordered fields
It is remarked in A Course in Model Theory: An Introduction to Contemporary Mathematical Logic by Bruno Poizat, page 101 (Google Books link), that the assertion is true for all ordered fields, but ...
- 50.6k
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