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23 votes

Hensel's proof that $e$ is transcendental

You can read Hensel's original article here. The argument is really simple: from this expansion, we see that $e$ satisfies an equation of the form $y^p=1+pu$ where $u$ is a $p$-adic unit. But the ...
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15 votes
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Centraliser of an absolute Galois group

For an extension $L/\mathbf{Q}_p$, let $G_L$ denote the absolute Galois group $\mathrm{Gal}(\overline{L}/L)$. If $\sigma \in G_{\mathbf{Q}_p}$ acts centrally on $G_K$, then it also acts centrally on ...
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15 votes
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Automorphisms of $\mathbb C_p$

If you strengthen the condition $\sigma(\mathbf Q_p) \subset \mathbf Q_p$ to $\sigma$ being the identity on $\mathbf Q_p$, so $\sigma$ is a $\mathbf Q_p$-automorphism of $\mathbf C_p$, then a simple ...
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14 votes
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Witt-vector vectors

It is impossible if you want reasonable functoriality, and for finite-dimensional vector spaces to be carried to finite free modules for $A$ a field. Presumably for $A = k$ a finite field of ...
13 votes
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Is the p-adic Lindemann-Weierstrass Conjecture still open?

Here is a 2018 paper, A Note on One-dimensional Varieties Over the Complex p-adic Field, that still lists the "full" statement as a conjecture; "half" of the statement, meaning that at least $\lfloor ...
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12 votes
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Zero of the exponential p-adic

The exponential function satisfies $\exp \left({x + y}\right) = \exp\left(x \right) \exp\left(y \right)$ for $x, y$ in the convergence domain. It also satisfies $\exp \left( 0 \right) = 1$. So if $\...
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  • 650
12 votes
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Question about log and exp of a formal group law

The radius of convergence of any formal group $F$ (one-dimensional, finite height) is $1$, in other words $L_F$ will converge at all $z\in\Bbb C_p$ with $v(z)>0$. In particular, the logarithm is ...
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  • 3,818
11 votes
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3-adic valuation of a sum involving binomial coefficients

The $3$-adic evaluation you seek is compactly given by $$\nu_3(a_{2n})=\nu_3\left(\binom{2n}n\right) \qquad \text{and} \qquad \nu_3(a_{2n+1})=\nu_3\left(3(2n+1)\binom{2n}n\right),$$ which can be ...
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10 votes
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In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?

Zero is the only algebraic integer which has all its conjugates strictly inside the complex unit circle. (Look at the norm.) For explicit examples with conjugates on either side of the unit circle, ...
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10 votes
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Convergence of a $p$-adic series

Is this question being asked just out of curiosity? As far as I know, the series $\sum_{n \geq 0} n!$ is not important in $p$-adic analysis. For $j \in \mathbf N$ and $q \not= 1$, let $(j)_q = (q^j-1)...
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10 votes
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P-adic functions on annuli

$\def\bQ{\mathbb{Q}}\def\bF{\mathbb{F}}\def\bZ{\mathbb{Z}}$This is false as stated because of the following important difference between $K$ and $\mathbb{C}$: the former is not algebraically closed. ...
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  • 6,517
10 votes

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

Let me answer your questions in the opposite order. (2) This question is vacuous: it cannot happen that both roots have valuation $> k-1$, because the product of the roots is $p^{k-1}$. So either ...
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9 votes

Witt-vector vectors

Let me restrict my attention to the only case I understand, which is $A = \mathbb{F}_p, W(A) = \mathbb{Z}_p$ (so here I mean $p$-typical Witt vectors). You don't provide any conditions you want your ...
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9 votes
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Algebraic numbers in all $\mathbb Q_p$

(cw answer based on Wojowu's link) The only algebraic extension of $\mathbf{Q}$ that embeds into $\mathbf{Q}_p$ for all $p$ (or even for a density 1 set of primes) is $\mathbf{Q}$ itself. For if $P\...
8 votes

Texts about Dwork's work

MR0498577 (58 #16672) Reviewed Katz, Nicholas Travaux de Dwork. (French. English summary) Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409, pp. 167–200. Lecture Notes in Math., Vol. 317, ...
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8 votes
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When does the radius of convergence of the product of two $p$-adic power series increase?

Here is a counterexample to your question at the end, for each $p$. Let $f_u(x) = x + ux^p/p$ for $u \in \mathbf C_p$ with $|u|_p = 1$ and $|u-1|_p = 1$. (Such $u$ can be taken in $\mathbf Z_p^\times$ ...
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8 votes

Zero of the exponential p-adic

It’s hard to see in what sense it could be true that the exponential function is “defined over $\Bbb C_p$”, since the logarithm is defined on the whole open unit disk there, and has so very many zeros....
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8 votes

Vector bundles on adic spaces

The question is local on $X$, so we may assume that $\mathcal E$ is finite free, of rank $n$, say. In that case, as also SashaP points out, the question amounts to the question whether $\mathbb A^n_X$ ...
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7 votes
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Vector bundles on the various sites of a preperfectoid

There is some relevant work of Ben Heuer on this. In short, analytic and etale vector bundles agree on all sousperfectoid adic spaces (and much more generally, for all "etale sheafy" adic ...
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6 votes

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

I can only give a very partial answer, and that only from my very parochial point of view. I will use the additive valuation $v$ rather than absolute value, normalized so that $v(p)=1$, and in terms ...
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  • 3,818
6 votes

3-adic valuation of a sum involving binomial coefficients

Indeed, the observation by Max Alexseyev seems to provide the idea I need! The conjecture I made above can be generalized as follows: Let p > 2 be a prime, and let $L_n (x)$ be the Legendre ...
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6 votes

The p-adic valuation of a linear recurrence

Actually, the binary linear recurrence case is pretty precise, especially if $p\ge3$ and you're working over $\mathbb Q$, and not over a field where $p$ is ramified. Let $r(p)$ denote the rank of ...
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6 votes

Automorphisms of $\mathbb C_p$

Transcendence degree ℂp/ℚp=2ℵ0 which is enough to answer the question affirmatively. More explicitly: ℂp embeds into the field of p-adic Malcev-Neumann series = formal sums &...
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6 votes

Is $K^\times/ F^\times$ compact for local fields?

$O_K^\times$ is compact thus so is $$K^\times/ \pi_F^\Bbb{Z}=\pi_K^{\Bbb{Z/eZ}} \times O_K^\times, \qquad e=\frac{v(\pi_F)}{v(\pi_K)}$$ Being a quotient of a compact group by a closed subgroup $$K^\...
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6 votes
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Vector bundles on adic spaces

$\newcommand{\cO}{\mathcal{O}}\newcommand{\bZ}{\mathbb{Z}}$Let's first work out the case $\mathcal{E}=\mathcal{O}_X$. We want a space $E\to X$ such that $Hom_X(S, E)=\cO_S(S)=Hom(S,\mathbb{A}^1)$. ...
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  • 6,517
4 votes
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Why is every l-adic Galois representation conjugate to one over the l-adic integers?

Recall that all ${\mathbb Z}_l$-lattices in the ${\mathbb Q}_l$-vector space ${\mathbb Q}_l^n$ are conjugate in ${\rm GL}_n ({\mathbb Q}_l )$. Since the ${\rm GL}_n ({\mathbb Q}_l )$ stabilizer of ...
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3 votes
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null infinite product in the p-adic setting

If $n_0$ is such that $|u_n|<1$ for $n \geq n_0$, then $|1+u_n| = 1$ for $n \geq n_0$, so the norm of the infinite product is the norm of the product of the first $n_0$ terms and hence nonzero.
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3 votes

In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?

To expand on GNiklasch's answer, and analyse what you write as well: we always have (when complex conjugation is central in the Galois group) have $\overline{\alpha^{\sigma}} = {\bar \alpha}^{\sigma}$ ...
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3 votes
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Definition of p-adic modular forms

It's not a pushforward of a differential (or maybe it can be described as one, but it doesn't have to be). Rather there is a pullback map from differentials on the elliptic curve to differentials on $...
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