The tag has no wiki summary.

learn more… | top users | synonyms

3
votes
1answer
146 views

Is there a multivariate analog of Dwork's theorem?

$f(x)=\sum_{i=0}^\infty a_ix^i$ with $a_i\in\Bbb Q$. Let $S$ be finite set of places of $\Bbb Q$ such that: 1. $\forall p\notin S$, $|a_i|_p\leq1$ $\forall i\geq0$. 2. $\forall v\in S$, $f(x)$ ...
9
votes
1answer
184 views

Theory of C* algebras over other fields than the complex numbers

How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same ...
7
votes
2answers
172 views

p-adic analogue of the Strong Law of Large Numbers

Is there a $p$-adic analogue of the Strong Law of Large Numbers? In particular, suppose that $f_i: \mathbb{Z}_p \longrightarrow \mathbb{Q}_p$ for $i = 1,2,\ldots$ is an sequence of random variables ...
4
votes
0answers
136 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 ...
5
votes
0answers
118 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 ...
3
votes
0answers
215 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
83 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
552 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
215 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
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 ...
2
votes
0answers
70 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
68 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
136 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
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
81 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
130 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
336 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
117 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
265 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
190 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
51 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
189 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
302 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
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
273 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
231 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
256 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
112 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
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
1answer
247 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
193 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
218 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
283 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
121 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
229 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
231 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
622 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
442 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
512 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
6k 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
858 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
244 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.
13
votes
3answers
2k 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
750 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
570 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
199 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
384 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 ...