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Questions tagged [p-adic]

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16
votes
2answers
1k views

Witt-vector vectors

I've never really made my way in any detail through the Witt-vector construction. I did read all the articles that a quick Google and MSN search turned up, and none seemed to address it, but I could ...
16
votes
2answers
471 views

3-adic valuation of a sum involving binomial coefficients

Let $$a(n) = \sum_{0 \leq k \leq n} {n \choose k}{{n+k} \choose k},$$ and define $b(n) = \nu_3 \bigl(a(n)\bigr)$, where $\nu_3$ is the $3$-adic valuation. About twenty years ago or so, I discovered (...
16
votes
1answer
373 views

Centraliser of an absolute Galois group

Let $K$ be a finite extension of $\mathbb{Q}_p$. Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ trivial ? If yes, how can I ...
15
votes
0answers
327 views

Power series which are $p$-adic modular forms for all $p$

Suppose that, for some integer $k$, a series $f(q) \in \mathbb Q \otimes \mathbb Z[[q]]$ has the property that for every prime $p$, $f(q)$ is the $q$-expansion of a $p$-adic modular form of weight $k$ ...
12
votes
0answers
411 views

The operator $\left(q\frac{d}{dq}\right)^s$ and fractional derivatives of modular forms

Recall the notion of a "nearly holomorphic modular form" introduced by Shimura: A function $f : \mathfrak h \to \mathbb C$ is said to be nearly holomorphic of level $\Gamma_1(N)$, weight $k$ and ...
12
votes
0answers
259 views

$p$-Adic or arithmetic variants of Khovanskii's “low complexity $\Rightarrow$ tame topology” theory

This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real ...
11
votes
1answer
590 views

Is the p-adic Lindemann-Weierstrass Conjecture still open?

The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$ be distinct $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\...
9
votes
1answer
581 views

Definition of p-adic modular forms

I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point. He first describes p-adic modular forms of tame level N as functions on the Igusa ...
8
votes
2answers
703 views

The p-adic valuation of a linear recurrence

Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely, $$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$ for some $a_1, \ldots, a_k \in \...
8
votes
1answer
265 views

How to deduce Bernstein-Zelevinsky classification from the Langlands one

I am trying to understand the Langlands classification. To that end, I am trying to find how I could deduce the Bernstein-Zelevinsky classifcation from the second description of the Langlands ...
8
votes
0answers
259 views

$p$-adic representations of the fundamental group of a smooth proper curve over a finite field

This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations $$ \pi_1(C)\...
7
votes
0answers
292 views

$p$-adic uniformisation of abelian varieties

In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement: Let $A$ over $\mathbf{Q}_p$ be an abelian variety ...
6
votes
2answers
448 views

Texts about Dwork's work

I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...
6
votes
1answer
526 views

p-adic L-functions of modular forms: why the condition $v_p(\alpha)<k-1$?

Let $f$ be a modular form (cuspidal, new, eigenform) of weight $k$ and level $N$. Let $p$ be a prime number not dividing $N$. In order to construct a $p$-adic $L$-function $L_p(f, s)$ interpolating ...
6
votes
1answer
591 views

Artin representations in Serre's book 'local fields'

Let $K$ be a complete local field with discrete valuation, and let $L/K$ be a finite Galois extension. Use $G=Gal(L/K)$ to denote the Galois group. In Serre's book 'local fields', chapter 6, a ...
6
votes
1answer
219 views

p-adic L-functions for (dual of) fine Selmer Groups

If $R(E/\Bbb Q_{\infty})$ is the fine Selmer group and $Y(E/\Bbb Q_{\infty})$ is its dual, then we know that $Y(E/\Bbb Q_{\infty})$ is a finitely generated $\Lambda$-module and by a theorem of Kato, ...
6
votes
0answers
254 views

What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
5
votes
2answers
492 views

When does the radius of convergence of the product of two $p$-adic power series increase?

Let $p$ be a prime number and denote by $R(f)$ the radius of convergence of a power series $f(x) \in \mathbb{C}_p[[x]]$, where $\mathbb{C}_p$ is the completion of the algebraic closure of $\mathbb{Q}...
5
votes
3answers
1k views

Is there any theorem like implicit function theorem in $\mathbb{Q}$ ?

My qeustion is that, is there any theorem like implicit function theorem in $\mathbb{Q}$ ? More precisely, let $p(\bar{x},\bar{y})$ be in $\mathbb{Z}[\bar{x},\bar{y}]$ such that in $\mathbb{Q}$, for ...
5
votes
1answer
138 views

$p$-adic valuation of $P_n(p)$ if $P_n$ is the $n$th Legendre polynomial

In 3-adic valuation of a sum involving binomial coefficients the $p$-adic valuation of $P_n(p)$ has been obtained. Computations indicate that $M_n(p)=\sum\binom{n}{k}^2(p-1)^k$ for odd $p$ has the ...
5
votes
0answers
110 views

Arithmetic of Cuspidal Reps. Fundamental non split stratum and simple stratum

I started to read Colin Bushnell's notes on this title. The last theorem in the 3rd section claims that if $\pi$ is an irr. smooth rep. of $GL_N(F)$ containing a non-split fundamental stratum, then $\...
5
votes
0answers
466 views

Equivalent definitions of the Hasse invariant

As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant. Let me start by recalling one definition: Let $E\to S$ be an elliptic curve in ...
4
votes
2answers
288 views

differences between character distributions of supercuspidal representations and others

Let $G$ be a $p$-adic linear reductive group. For an irreducible admissible smooth representation $\pi$ of $G$, there is a distribution $\Theta(\pi)$, called the character distribution, attached to $\...
4
votes
1answer
320 views

computing spaces of $p$-adic modular forms

Let $p$ be a prime, and $\alpha$ a positive integer. How do you compute the space of $p$-ordinary $p$-adic modular forms (in the sense of Serre) of weight 2 on $\Gamma_0(p^\alpha)$? I'm really only ...
4
votes
0answers
344 views

p-adic etale cohomology

Let $X$ be a smooth projective scheme over $\mathbb{Z}_p$, with special fiber $X_s$ over $\mathbb{F}_p$, generic fiber $X_{\eta}$ over $\mathbb{Q}_p$, and geometric generic fiber $\bar{X_{\eta}}$ over ...
3
votes
2answers
735 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 ...
3
votes
0answers
252 views

Cyclotomic Extension of a Perfectoid Space

Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more ...
2
votes
1answer
386 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 ...
2
votes
1answer
263 views

differential structures on unit-root Frobenius modules.

Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n, \; a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is ...
2
votes
1answer
594 views

History of Fargues-Fontaine curve

In this paper, Pierre Colmez wrote about some history of the Fargue-Fontaine curve. In this schedule of London Number Theory Study Group, Fargues was said to give a talk on November 15th on " Where ...
2
votes
1answer
170 views

“frequency” of fields for which the p-adic regulator vanishes (mod p)

There is a very nice question which arises in the study of the Discrete Logarithm Problem which I wish to present here. The question, in a general setting, is to specify an empirical expression for ...
2
votes
1answer
253 views

Can one determining the p-adic lattice just from the values of the quadratic form on a p-group?

Given a finite $p$-group $A$, with a non-degenerate quadratic form $q:A\rightarrow \mathbb Q/2\mathbb Z$ (that is a map satisfying $q(na)=n^2q(a)$ for all $n\in \mathbb Z,a\in A$), an important result ...
2
votes
0answers
61 views

Invariant compact in division ring

Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in ...
2
votes
0answers
210 views

Control theory for Kitagawa's $\Lambda$-adic modular symbols

Let $p$ be a prime, $\Gamma=1+p\mathbb Z_p$ and $\Lambda=\mathbb Z_p[[\Gamma]]$ the Iwasawa algebra. Let $\kappa\colon\Gamma\rightarrow\mathbb Z_p^\times$ be the inclusion. For a character $\...
2
votes
0answers
302 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
172 views

p-adic dual spaces [closed]

I am trying to determine some properties of Lipschitz distributions. To do so, I need to know the dual space for $l^\infty$. The sequences tending to zero are certainly in the dual space to $l^\...
1
vote
1answer
104 views

null infinite product in the p-adic setting

Let $p$ be a prime, $\mathbb C_p$ be the completion of a algebraic closure of $\mathbb Q_p$ and $(u_n)_{n\in\mathbb N}$ be a sequence of $\mathbb C_p$ converging towards $0$. Suppose that for all $n\...
1
vote
1answer
477 views

Why is every l-adic Galois representation conjugate to one over the l-adic integers? [closed]

Why is every l-adic Galois representation $$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Q}_{l})$$ conjugate to one over the l-adic integers? $$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$
1
vote
1answer
232 views

Specialisation of Z_p[[S_1,…,S_n,X_1,…,X_d]]

Let $R \colon= {\Bbb Z}_p[[S_1,...,S_n,X_1,...,X_d]]$ be the $(n+d)$-variable formal power series ring over ${\Bbb Z}_p$. Choose an arbitrary ideal ${\frak b}$ of $R$ such that ${\mathrm{ht}}({\...
1
vote
1answer
248 views

unramified base change in characteristic p > 0?

Hi, Suppose that $E/F$ is a unramified extension of local fields of characteristic zero. Let $G = GL_n$. Then it is well-known (due to Clozel?) that base change of tempered representations from $G(F)$...
1
vote
0answers
55 views

Automorphisms in function fields with $\mathbb F_q[T]$ globally invariant

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q((1/T))$ for the valuation $-\deg$. Does there exist uncountably many automorphisms of $\Omega$ that let $\mathbb F_q[T]$ stable ...
1
vote
0answers
120 views

Crystalline cohomology over arbitrary perfect ground fields

In preparing a set of notes, I realize there is a simple question for which I cannot find an easy answer; likely due to simple ignorance. Any introductory text on etale cohomology notes that ...
1
vote
0answers
147 views

Kitagawa's p-adic modular symbols for different weights: a confusing observation

References are to K. Kitagawa, "On standard $p$-adic $L$-functions of families of elliptic cusp forms", Contemp. Math. 165, 1994. Let $\mathcal O$ be the ring of integers in a finite extension of $\...
1
vote
0answers
68 views

Dimensions of fibers of analytic map

I must admit that I know nothing about p-adic geometry, so the following question may be completely trivial. Let $V\subset K^n$ be an affine algebraic variety. Let $D$ be a polydisk, and $F$ be an ...
0
votes
1answer
119 views

Twisted Padé approximants

Let $f$ be a continuous function defined on $\mathbb Z_p$. By Mahler theorem, there exists a sequence $(\gamma_k)_{k\in\mathbb N}$ of $\mathbb Z_p$ such that for every $z\in\mathbb Z_p$ $$f(z)=\sum_{k\...
0
votes
0answers
114 views

How is the p-adic norm calculated when using universal witt vectors?

How is the p-adic norm calculated when using UNIVERSAL WITT VECTORS? Is the p-adic norm calculated in the familiar way, in the sense that we look to the last digit to the right, and the prime number ...
0
votes
0answers
192 views

Name of some commutative ring akin to $p$-adics

I need help in identifying the naming convention of some commutative ring described below. Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...
0
votes
0answers
222 views

Let $u \in \mathbb{Z}^{\times}_{p}$ a unit, is it true that the field $\mathbb{Q}_{p}(\zeta_{p^{n}}, \sqrt[p^{n}]{u})$ is abelian over $\mathbb{Q}_{p}$?

Let $u \in \mathbb{Z}^{\times}_{p}$ a unit, is it true that the field $\mathbb{Q}_{p}(\zeta_{p^{n}}, \sqrt[p^{n}]{u})$ ` is abelian over $\mathbb{Q}_{p}$?