The p-adic tag has no usage guidance.

**1**

vote

**0**answers

40 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

**0**answers

63 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

**0**answers

154 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 ...

**1**

vote

**1**answer

260 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?
$$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$

**9**

votes

**0**answers

141 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 ...

**7**

votes

**1**answer

267 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 ...

**7**

votes

**1**answer

289 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 ...

**12**

votes

**0**answers

170 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 ...

**0**

votes

**0**answers

66 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**

votes

**1**answer

101 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

**1**answer

154 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**

votes

**2**answers

383 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 ...

**12**

votes

**0**answers

260 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 $k$ ...

**3**

votes

**2**answers

593 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

**0**answers

189 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

**1**answer

322 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**

votes

**1**answer

122 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**

votes

**1**answer

257 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**

votes

**2**answers

170 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

**1**answer

377 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

**2**answers

157 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**

votes

**1**answer

261 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

**0**answers

206 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

**1**answer

222 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**

votes

**0**answers

137 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**

votes

**3**answers

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 ...

**2**

votes

**1**answer

209 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 ...