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14
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0answers
1k views

The radius of convergence of the p-adic exponential function.

As every number theorist learns, the radius of convergence of $exp(x)$, defined by the usual power series in a neighborhood of zero, is $$\rho = p^{-1/(p-1)}.$$ This is typically proven by computing ...
7
votes
0answers
275 views

Differences in tree picture of ${\bf Q}_p$, $\overline{{\bf Q}_p}$, ${\bf C}_p$, $\Omega_p$

I was discussing the tree picture of ${\bf Z}_p$ and ${\bf Q}_p$ and mentioned that the idea can be extended to ${\bf C}_p$, with the caveat that the tree is no longer locally finite (as the value ...
5
votes
0answers
116 views

Non-embeddable varieties

Suppose that $k$ is a perfect field of characteristic $p>0$, $\mathcal{V}$ is a complete discrete valuation ring with residue field $k$ and quotient field $K$, of characteristic $0$. Then when ...
5
votes
0answers
135 views

The topology on the Robba ring

I've been reading Kedlaya's paper http://arxiv.org/abs/math/0208027 on finiteness of rigid cohomology and there's something I can't quite resolve in my understanding of the topology on the Robba ring. ...
5
votes
0answers
230 views

2-adic Logarithm and Resistance of n-dimensional Cube

Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is $$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=1}^{n}\frac1{{n-1\choose k}}.$$ The ...
5
votes
0answers
450 views

a naive question about p-adic local monodromy theorem

The question is about whether one can view the p-adic local monodromy theorem as the quasi-unipotence of some monodromy operator. it is known that the classical local monodromy theorem (i.e. for ...
5
votes
0answers
210 views

Lower bound for p-adic distance between roots

Let $f$ be a formal power series with coefficients in the ring of integers of a finite extension of ${\mathbb Q}_p$. Is there a simple algorithm to compute a positive lower bound for $|\alpha - ...
4
votes
0answers
133 views

Simultaneously using the real and 2adic norms

In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to ...
4
votes
0answers
116 views

Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...
4
votes
0answers
188 views

irrationality of the p-adic exponential

I would like to illustrate my lecture on p-adic numbers with some elementary results. I proved that the series $e=\sum_{n\ge0}\frac{p^n}{n!}$ converges in $\mathbb Q_p$ for every prime $p$. Now I ...
4
votes
0answers
508 views

Newton Method in $p$-adic case

The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentialbe assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic ...
3
votes
0answers
212 views

Flatness over a perfectoid ring

I want to prove the following: Let $R$ be a perfectoid ring and $\varpi$ a pseudo uniformizer in $R$ which admits all $p$-th power roots, then a module over $R^\circ$ is flat if and only if it has no ...
3
votes
0answers
82 views

Is there a Noetherian profinite group of infinite rank?

Is there a profinite group $G$ such that any closed subgroup $H \leq G$ is finitely generated, but there is no $n \in \mathbb{N}$ such that every closed subgroup of $G$ can be generated by at most $n$ ...
2
votes
0answers
69 views

Weil index computation, p-adic integral

The following peculiar p-adic integrals have arisen in my work, and I would be interested if anyone can see how to tackle them. Let $F$ be a $p$-adic field, $\mathfrak{o}$ its ring of integers, ...
2
votes
0answers
50 views

Morita equivalence for utrametric Banach algebras (reference needed)

Is there any descent description of Morita theory for ultrametric Banach algebras? To make this question more precise let $K$ be some completion of the field $\mathbb{Q}_p$ (I'm mostly interested in ...
2
votes
0answers
120 views

A nice rigid analytic model for local systems over an elliptic curve?

For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts ...
2
votes
0answers
294 views

reference for p-adic Stein spaces

Hi, I'm looking for a reference in english for p-adic Stein spaces. The usual referneces I come across are all in german. Thanks
1
vote
0answers
192 views

p-adic Lie theory

It is well known that exponential map in $C^{n\times n}$ will cover all non-sigular matrix $GL(n,C)$, which is a basic fact in Lie group and lie algebra theory, whether it is true for p-adic cases. ...
0
votes
0answers
72 views

Nilpotent differential operators

I am reading Dwork's book an Introduction to G-Funcions and confronted with a problem. In section 2, chapter III (page 81), he assumes that $\mathscr F=K(X)$ the field of rational functions with ...
0
votes
0answers
90 views

Computing a projection of a $p$-adic plane curve

Given a prime $p$ and a polynomial equation $f(x,y)=0$ with rational coefficients, I would like to obtain a precise description of the set of all numbers $y\in\mathbb Q_p$ such that the equation has a ...
0
votes
0answers
79 views

uniform continuity of a function in ultrametric spaces

Consider $[0,1]$ with the metric $d_1(x,y)=\left\{\begin{array}{cc} 0&x=y,\\ \max\{x,y\}&x\ne y. \end{array}\right.$. Moreover let $(M,d_2)$ be an ultrametric space. Let ...
0
votes
0answers
181 views

limit of $p$-adic polynomials

Let $Q_n$ be a sequence of polynomials of $\mathbb C_p(x)$ such that for every $z\in\mathbb C_p$ we have $\lim_{n\to+\infty}Q_n(z)=0$. One assumes that for every $n\in\mathbb N$, there exist two ...
0
votes
0answers
264 views

What is $p$-adic Fourier series?

Q1: Can we define Fourier series for a function $\mathbb{Z}_p\to \mathbb{Q}_p$? Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion: ...
0
votes
0answers
118 views

the definition of pro-infinitesimal thickenings

Let $R$ be a ring and $A$ and $R$-algebra. A pro-infinitesimal thickening of $R$ is a pair $(D, \theta)$ such that $\theta: D \rightarrow R$ is surjective and $D$ is separated and complete for the ...
0
votes
0answers
265 views

Does it exist a p-adic L function which interpolates the values of the complex one at positive integers?

I known that there are classical ways to construct $p$-adic $L$ functions for Dirichlet characters through $p$-adic integrals. We fix a character $\chi$ modulo $Np^r$ with $N$ and $p$ coprime and an ...
-2
votes
0answers
45 views

how to calculate the radius of convergence of the p-exponentials of Pulita?

please it is known from the Pulita thesis that the radius of convergence of his pi-exponentials is 1; he used a differential operator which has this pi-exponetial as a solution etc..., but me I ...