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0
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0answers
45 views

Explicit twisted Padé approximants

This is a follow-up of Twisted Padé approximants Let $z\in\mathbb Z_p$ with $v_p(z)>0$. One puts $f_z(x)=(1+z)^x$ for all $x\in\mathbb Z_p$. I try to determine the twisted Pade approximants ...
0
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1answer
96 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 $z\in\mathbb Z_p$ ...
2
votes
1answer
135 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 ...
6
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2answers
359 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 ...
11
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0answers
189 views

Power series which are $p$-adic modular forms for all $p$; a local-to-global principle?

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 $k$ ...
3
votes
2answers
556 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 ...
2
votes
0answers
181 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 ...
5
votes
1answer
276 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 ...
2
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1answer
103 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 ...
4
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1answer
252 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 ...
3
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2answers
159 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
327 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 ...
1
vote
2answers
156 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 ...
2
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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
1answer
217 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 ...
0
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0answers
125 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 $\mathbb{Q}_{p}$?
5
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3answers
963 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 ...
2
votes
1answer
186 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 ...