The p-adic tag has no usage guidance.

**3**

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**1**answer

63 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 ...

**13**

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266 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**

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**0**answers

181 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 ...

**9**

votes

**0**answers

149 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 ...

**2**

votes

**0**answers

123 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**

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193 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 ...

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vote

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48 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 ...

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vote

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211 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.
...

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votes

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78 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 ...

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votes

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172 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 ...

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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**

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141 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 $\mathbb{Q}_{p}$?