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 ...
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27 votes
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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'...
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20 votes
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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 ...
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11 votes
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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 ...
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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 ...
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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
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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 $2\pi i\mathbb Q$. ...
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  • 9,510
7 votes
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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. ...
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6 votes
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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 ...
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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 ...
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6 votes
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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.
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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 ...
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  • 1,253
5 votes
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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 ...
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5 votes
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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 ...
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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 ...
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5 votes
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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 ...
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5 votes
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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 ...
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  • 7,931
5 votes
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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 ...
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5 votes
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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 ...
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  • 23.7k
4 votes
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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 ...
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  • 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.
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  • 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, ...
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  • 1,321
4 votes
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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 ...
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3 votes
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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: ...
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3 votes
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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 ...
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3 votes
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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 ...
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  • 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 ...
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  • 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{\...
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3 votes
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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 $\varphi(x)$ ...
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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 ...
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  • 50.6k

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