Tag Info

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
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
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
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

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

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 ...
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
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
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

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
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

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
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
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

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
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
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
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
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
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

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

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
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
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
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
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

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
Accepted

• 1,454