The p-adic-analysis tag has no wiki summary.
3
votes
0answers
77 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 $n$ ...
3
votes
2answers
516 views
Automorphisms of $\mathbb C_p$
I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.
If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that ...
7
votes
1answer
212 views
p-adic Stein spaces
The higher cohomology of coherent sheaves vanish on Stein spaces (both complex and p-adic). In the case when the space ($X$) is a curve and we're working in the complex world, this shows that all ...
0
votes
0answers
71 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 ...
2
votes
0answers
67 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, ...
1
vote
1answer
66 views
Slope decomposition of a product of operators
I'm trying to relate the slope decomposition of a product of linear operators to the slope decompositions with regard to each of the operators in the product.
First I'll give some background, for ...
5
votes
0answers
121 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.
...
0
votes
0answers
88 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
78 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
...
1
vote
1answer
127 views
$p$-adic Regulators
Is there some relationship between the $p$-adic regulators of isogenous curves over $\mathbb{Q}$? I've done some computations and their ratio seems to be related (equivalent in all calculations so ...
8
votes
1answer
301 views
is there a p-adic implicit function theorem?
I am trying to find a good reference for a version of the implicit function theorem over $p$-adic manifolds. None of the texts I have consulted ( including "$p$-adic numbers, $p$-adic analysis, and ...
4
votes
0answers
112 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
1answer
249 views
Iwasawa logarithm and analytic continuation
I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$.
...
2
votes
0answers
171 views
Transcendental numbers in the p-adic rationals $\mathbb Q_p$ [closed]
I know that there are uncountably infinite transcendentals over $\mathbb Q$ in $\mathbb Q_p$. What i want to ask is if there is a way to determine whether a transcendental over $\mathbb Q$ lays in ...
2
votes
0answers
48 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 ...
4
votes
0answers
178 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 ...
6
votes
1answer
292 views
What are the automorphisms of a perfectoid Tate algebra?
Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a_1T+a_2T^2+\cdots$, where $a_1\in ...
0
votes
0answers
179 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
258 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:
...
5
votes
0answers
216 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 ...
4
votes
2answers
254 views
Proving the existence of an integral quadratic form
Theorem 11 (Conway & Sloane, Sphere Packings, Lattices and Groups, 3rd Edition, pp 383, Ch 15). If a system of putative $p$-adic symbols for each $p$ satisfies the determinant, oddity, and ...
1
vote
1answer
111 views
How to understand the infraconnected set and affinoid?
I begin to study some p-adic analysis. I find it is hard to understand the infraconnected set and affinoid. It is strange that I cannot find them at wiki and only a few book(by the same auther) ...
2
votes
0answers
119 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
1answer
239 views
Trivial p-adic measures
I am looking at p-adic distributions, and in this case p-adic measures. To say that $\mu$ is a distribution means that the arguments of $\mu$ are compact open subsets of $\mathbb{Z}_p$, $\mu$ is ...
1
vote
0answers
188 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.
...
1
vote
2answers
205 views
Spectrum theorem for p-adic matrix analysis
Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way:
1: whether there exist spectrum theorem for p-adic matrix.\
2: ...
7
votes
0answers
268 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 ...
0
votes
0answers
117 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 ...
7
votes
1answer
227 views
Describing the ratio of uniformizers in B_dR
In Conrad and Brinon's notes http://math.stanford.edu/~conrad/papers/notes.pdf, two uniformizers of $B_{dR}$ are produced: one is $\xi := [\tilde{p}]-p$ (bottom of p.58), where $\tilde{p} = (p, ...
5
votes
1answer
228 views
What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$?
I am looking for a description of the Plancherel Measure of $\textrm{SL}_3(\mathbb{Q}_p)$. Has this been calculated yet? I've search many places for it, but I've only found results on real/complex ...
-4
votes
3answers
605 views
Books on analytic functions on Banach spaces over a non-Archimedean field
I'm looking for good textbooks on analytic functions on Banach spaces over a non-Archimedean field.
If you know one(s), please let me know.
6
votes
1answer
431 views
A p-adic analogue for a formula of Riemann?
This might be naive question but I was wondering whether a p-adic analogue of the following (shockingly) beautiful formula $$\zeta(s)\Gamma(s) = \int_0^\infty \frac{t^{s-1}}{e^t-1} dt$$ (vaild for ...
4
votes
0answers
488 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 ...
1
vote
1answer
190 views
The set of $p$-adic numbers of some fixed $n^ {\rm th}$-power residue
Hi,
Consider the set $\lambda P_n \subset \mathbb{Q}_p$, where $\lambda \in \mathbb{Q}_p^{\times}$ and $P_n$ is the set $\lbrace x \in \mathbb{Q}_p \mid \exists y \in \mathbb{Q}_p x= y^n\rbrace$. Is ...
16
votes
11answers
3k views
'Important' applications of p-adic numbers outside of algebra (and number theory).
Surely, $\mathbb{Z}_p$ and $\mathbb{Q}_p$ (and their extensions) are very important for algebra and number theory. Do they have any important applications outside of algebra (that I could easily ...
38
votes
10answers
5k views
Elementary results with p-adic numbers
I'm giving a talk for the seminar of the PhD students of my math departement. I actually work on Berkovich spaces and arithmetic geometry but, of course, I cannot really talk about that to an audience ...
7
votes
2answers
854 views
P-adic representations
Hi,
I am reading about p-adic representations from Fontaine's book which can be found at http://staff.ustc.edu.cn/~yiouyang/research.html. On page 145
where they prove Proposition 5.24 which is ...
1
vote
1answer
237 views
Partitioning a compact open set into balls in an ultrametric space
Consider a $p$-adic field $K$ with the standard topology inherited from the usual $p$-adic norm $\mid \cdot \mid$. Consider the ultrametric space $X=K^n$ with the topology inherited from the norm $\| ...
2
votes
1answer
314 views
Terminology-history of p-adic representations
Where appears for the first time the term Hodge-Tate representation.
Can i find somewhere explanation of the terminology Hodge-Tate, Derham etc. for representations and Fontaine's rings.
12
votes
3answers
1k views
When are roots of power series algebraic?
Let $K$ be a field and consider a power series $f(T) \in K[[T]]$. Under what conditions (on $K$ and/or on $f$) can we conclude that if $\alpha$ is a root of $f(T)$ then $\alpha$ is in fact ...
13
votes
1answer
739 views
P-adic C* algebras
I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of ...
10
votes
1answer
560 views
Can local duality for elliptic curves be proven with “big rings”?
From Exercise 5.14, Ch. V of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", I learned that the local duality for elliptic curves over $p$-adic fields can be proven for Tate curves ...
1
vote
1answer
198 views
Is anything known about flow along vector fields over complete normed fields?
In Bourbaki "Variétés différentielles et analytiques" there is a statement (without proof) that for a vector field on smooth manifold over complete normed field (characteristic zero) there is a local ...
5
votes
1answer
380 views
Continuous extensions reals and to p-adic numbers
Assume $f\colon \mathbb Q\to \mathbb Q$ is a function which admits continuous extensions
$f_0\colon\mathbb R\to \mathbb R$ and
$f_p\colon \mathbb Q_p\to \mathbb Q_p$ for each prime $p$.
Is ...
5
votes
0answers
441 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
209 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 - ...
18
votes
1answer
2k views
Field with one element example?
$$\frac{1}{\mu(B)}\int_B \vert x \vert d\mu(x) = \frac{1}{p+1}$$
This formula holds for the unit ball in $\mathbb{Q_p}$. This formula also holds for
$\mathbb{R}$ when $p=1$. Should one expect ...
0
votes
1answer
357 views
Relating $p$-adic Valuations of Elements in $\mathbb{C}$ and $\mathbb{C}_p$
Let $K = \mathbb{Q}(\theta)$, where $\theta$ is a root of an irreducible polynomial $g \in \mathbb{Z}[t]$. Fix a rational prime $p$. Let $\theta^{(1)}, \ldots, \theta^{(n)}$ be the roots of $g$ in ...
7
votes
1answer
492 views
The Galois representation of a p-divisible group is crystalline
Can someone explain (or give a reference) why the Galois representation attached to a p-divisible group over the ring of integers of a p-adic ring is Crystalline?
2
votes
0answers
290 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
