# Questions tagged [p-adic-analysis]

p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.

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### Does maximally incompleteness cause nonvanishing of the extension of maximal ideal of a valuation ring by rank 1 free module?

In B. Bhatt's lecture notes[1], Remark 4.2.5 says
... $\operatorname{Ext}_R^2(k,R)$ is non-zero if $K$ is not spherically complete.
which amounts to the following pure algebraic question.
Statement ...

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### Classification of submultiplicative ring norms on $\mathbb Q$

Let $R$ be a ring with identity. I call a non-negative real valued function $N: R \to \mathbb R_{\geq 0}$ a ring norm, if it has the following properties:
$N(r) = 0$ iff $r = 0$
$N(r+s) \leq N(r) + N(...

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### Does $P(\exp_p(a),\exp_p(b))=0$ imply $P=0$, where $\exp_p(\cdot)$ is $p$-adic exponential?

Classical case:
Let $\{a,b\}$ be linearly independent set over $\mathbb Q$ and $\{e^{at},e^{bt}\}$ be linearly independent set over $\mathbb Q[[t]]$. Suppose $P(x,y)$ is a polynomial over $\mathbb Q$. ...

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### When is the power-bounded subring top. of finite type?

Very naive question here. Let $K$ be a complete nonarchimedean field, $A$ a reduced affinoid $K$-algebra. When is the power-bounded subring $A^\circ$ topologically of finite type, in the sense that we ...

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### Constituents of $C_0^\infty(F^\times)$ for the regular action

Let $F$ be a $p$-adic field, and $C_0^\infty(F^\times)$ the space of smooth compactly supported functions on $F^\times$. Under the regular action of $F^\times$ on $C_0^\infty(F^\times)$, I believe we ...

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### $\lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p$ for any $a \in \mathbb{Z}^+$?

$\newcommand\tetra[2]{{^{#1}{#2}}}$In a recent discussion on the Tetration Forum (see https://math.eretrandre.org/tetrationforum/showthread.php?tid=1703&page=2), it has been pointed out how my ...

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### Is insoluble $p$-adic analytic just-infinite pro-$p$ group torsion-free?

Recall that an infinite pro-$p$ group $G$ is called just-infinite if all non-trivial closed normal subgroup of $G$ have finite index.
Question: Let $G$ be an insoluble $p$-adic analytic just-infinite ...

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### $p$-adic analogue of modular forms, upper half-plane, and $L$-functions

In the classical picture, there is the (complex) modular form, defined on the (complex) upper half plane, which is related to the (complex) $L$-function via the Mellin transform. As I have recently ...

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### Quotients of pro-$p$ groups linear over a complete Noetherian local ring

Let $R$ be complete Noetherian local ring with finite residue field $\mathbb{F}$ of characteristic $ p $. We say that a pro-$p$ group $G$ is linear over $R$ if it is isomorphic to a closed subgroup of ...

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### p-adic period map in Lawrence and Venkatesh

In Lawrence and Venkatesh's paper on the Mordell conjecture, they prove that there are finitely many $K$-rational points on a hyperbolic curve $X$, where $K$ is a number field, by showing that there ...

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### Does a $p$-adic power series $F(x,y)=\sum_{i,j \geq 0}b_{ij}x^iy^j \in \mathbb Z_p[[x,y]]$ have finitely many zeros in $\mathfrak{m}_{\mathbb C_p}$?

Let us consider the $p$-adic field $\mathbb Q_p$ with ring of integers $\mathbb Z_p$ and maximal ideal $\mathfrak{m}$.
Then any $p$-adic power series $f(x)=\sum_{n>0}a_nx^n \in \mathbb Z_p[[x]]$ ...

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### Is the ring of power series with $p$-adic coefficients Huber?

I have been reading the Berkeley lectures and got stuck with this question. Let $\mathbb{Q}_p [[t]]$ denote the ring of power series with $p$-adic coefficients. Is there a natural topology (e.g. the ...

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### The dimension of a torsion-free $p$-adic analytic group generated by two generators

$\DeclareMathOperator\GL{GL}$Let $G$ be a $2$-generator pro-$p$-group of finite rank, i.e. it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some integer $d$. Assume that $G$ is ...

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### Why $p$-adic measures?

I'm currently learning about the Kubota–Leopoldt $p$-adic $L$-function and I'm noticing that many people view the Kubota–Leopoldt $p$-adic $L$-function as a measure as opposed to a $p$-adic analytic ...

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### On the noetherianess of some subalgebras of an affinoid algebra

$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...

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### Interpretation of $p$-adic 'smoothness'

real case: In the very first course of Calculus, one learns that a real function $f \colon \mathbb{R} \to \mathbb{R}$ is called smooth, if it is differentiable as many times as one pleases. So the ...

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### Composition of p-adic power series

I asked a similar question on math stack exchange, but I figured it would be better to post here too.
Here's the link to the original post.
Take $K$ to be a finite (complete) extension of $\mathbb{Q}...

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### Criterion to decide whether a function is algebraic

For Christol's theorem see 1, if a power series is algebraic over every fields of characteristics $p$, is it algebraic over fields of $0$, by Robinson's principle?
Update: The power series is in $\...

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### A non-$p$-adic proof of a congruence of Bernoulli numbers

In A Multimodular Algorithm for Computing Bernoulli Numbers, Harvey uses the following congruence for Bernoulli numbers:
$$B_k \equiv \frac{k}{1-c^k} \sum_{x=1}^{p-1} x^{k-1} h_c(x)\quad(\text{mod}\ p)...

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### How to plot a p-adic function? [closed]

I found on the Internet some ways to provide a graphical representation of the $p$-adic integers or numbers (e.g., these illustrations of Heiko Knospe). They all exploit the fact that $p$-adic ...

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### An application of Koike's Trace Formula

Koike's Trace Formula states that
\begin{equation}
\mbox{Tr}((U_p^{\kappa})^n) = - \sum_{0 \leq u < \sqrt{p^n}\\
(u,p)=1}H(u^2-4p^n)\frac{\gamma(u)^\kappa}{\gamma(u)^2 - p^n}-1,
\end{equation}
...

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### Has any one seen this sum of roots of unity before?

Fix a prime $p >2$ and $q_1$, $q_2$ such that $q_i - 1$ is exactly divisible by $p$. For any $n$, $a$, $b $, consider the sum
$$\sum_{i=0}^{p^{n-1}-1}\zeta_{p^n}^{aq_1^i+bq_2^i}.$$
Is this always ...

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### question about Sinnott's proof of the Ferrero-Washington Theorem

I'm currently reading the paper "On the $\mu$-invariant of the Γ-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian number ...

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### Analogs of the Weil conjectures for non-archimedian fields

Suppose that $X$ is a smooth and proper variety defined over a perfect non-archimedian valued field $k$ of characteristic $p$. Then one can consider the action of Frobenius on crystalline cohomology. ...

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### The Gamma-transform and $p$-adic $L$-functions

I'm currently reading the paper "On the $\mu$-invariant of the $\Gamma$-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian ...

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### Geometric series in algebraic number fields

For which algebraic numbers $\alpha$ is there a valuation on the number field ${\mathbb {Q}}(\alpha)$ for which the infinite series $\sum_{n=0}^\infty \alpha^n$ converges to $1/(1-\alpha)$?

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### p-adic taylor polynomial [closed]

This might be an easy question but i am sorry for asking this.
Let $f(x)\in\mathbb{Z}_p[x].$ Is it always true that
$$f(x+y)=f(x)+f'(x)y+f''(x)\frac{y^2}{2}+zy^3$$
for some $z\in\mathbb{Z}_p.$ if it ...

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### Hensel's proof that $e$ is transcendental

When he introduced $p$-adic numbers, Kurt Hensel produced an incorrect local/global proof of the fact that $e$ is transcendental. Apparently, the intended proof goes along the following lines: ...

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### Elementary aspects of The Fargues-Fontaine curve

To any pair $(E,F)$, where $E$ is a local field and $F$ is a perfectoid field, one can associate a curve $X^{\text{FF}}_{E,F}$, the so-called Fargues-Fontaine curve, which is unique up to Frobenius ...

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### Evaluating $\sum_{n=0}^\infty n^k n!$ in p-adics, and its connection to the summation of divergent series

Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...

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### Key ideas behind p-adic Baker's theorem

I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...

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### In need of help with parsing non-Archimedean function theory

My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I'...

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### Decomposition of primes in cyclotomic extensions and their ramifications

Let $p$ be a prime. Suppose $L$ is a degree $p$ Galois extension over a number field $K$. Suppose $p$ splits both in $K$ and $L$.
So there will be $[K:\mathbb{Q}]$ primes of $K$ over $p$. Call them $...

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### Difficulty about Jordan decomposition, (and also an ambiguity about the quadratic forms in indecomposable Jordan components of quadratic modules)

I am trying to understand a concept through solving some exercises, but I can't solve one of them, and I need a hint and guide.
I asked my questions in the boxes (See the end of this question). (I ...

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### Transcendentality of Coleman integral

I wonder if there's any work that considers the algebracity/transcendentality of Coleman integral (over $\mathbb{Q}_{p}$). The reason I think about this is because, for hyperelliptic curves, there are ...

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### What is the preimage of the maximal ideal under certain exponential functions?

I'm taking a shot in the dark with this question, so I apologize if it makes no sense.
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $K_n$ be the field obtained by adjoining the $n$-th ...

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### p-adic density of the image of a polynomial

Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. Recently, a user of MO proved that the limit
$$\delta_p(P) := \lim_{n \to \infty} \frac{|\{P(x) \bmod p^n : x = 1,\...

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### Existence of "nth root function" which is analytic

Let $K$ be a finite extension of $Q_p$. Let $q$ be the size of the residue field of $K$, and let $\pi$ be a uniformizer of $K$. Then $q/\pi$ is some power of $\pi$ up to a unit $u$ in $K$, say $q/\pi =...

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### Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?

When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...

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### Bernoulli distributions and $p$-adic measure on $K$

The $p$-adic field $\mathbb{Q}_p$ has topological basis of open sets of the form $a+p^N \mathbb{Z}_p$ for $0 \leq a \leq p^N-1$ and $N \in \mathbb{Z}$. These are indeed compact open sets. One can ...

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### A p adic limit of a binomial coefficient

Let $0 \leq a \leq p^n$ be a number coprime to p. Consider the following sequence of binomial coefficients:
$$B_k = \binom{p^{n+k}}{p^ka} $$
as $k\to \infty$. If I did the computation right, the p-...

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### Local global principle for a system of polynomial equations

Suppose $T$ be a system of polynomials homogenous of degree 2 solvable over $\mathbb{R}$ and $\mathbb{Q}_p$ for all primes $p$. So, can we claim that $T$ is solvable over $\mathbb{Q}$? I think as of ...

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### Can perfect numbers be seen $p$-adically?

It is well known that all even perfect numbers are of the form $N=(2^{q}-1).2^{q-1}$ with $M_{q}:=2^{q}-1$ a Mersenne prime.
As the very defining property of such a perfect number is to fulfill the ...

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### Computing the ring of power-bounded elements in an affinoid algebra

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $A$ be an affinoid $K$-algebra, i.e. $A$ is isomorphic to a quotient of the Tate algebra $K\left<T_1,\dotsc,T_n\right>$ for some $n$. ...

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### Extension of hyperderivatives

Let $a$ be algebraic over $K:=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. Can one extend continuously the hyperderivatives on $K(a)$?
Recal that the hyperderivative $D_h$ over $K$ is defined by
...

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### $C^*$-algebras over an extension of $\mathbb{Q}_p$?

I'm wondering to what extent it might be possible for the theory of $C^*$-algebras to be translated into the $p$-adic context i.e. to define 'p-adic $C^*$-algebras' over some extension of $\mathbb{Q}...

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### 'Spherically complete' normed fields

A non-Archimedean normed field $K$ is said to be spherically complete if every decreasing sequence of closed balls in $K$ has non-empty intersection. I am a little puzzled as to why this definition is ...

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### Automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers?

I am interested to see what is currently known about the automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers (with respect to the $p$-adic topology induced by the $p$-...

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### A "multi-adic" absolute value / topology?

Let $S$ be a set of finitely many prime numbers. Then, define $\left|\cdot\right|_{S}:\mathbb{Q}\rightarrow\left[0,\infty\right)$ by: $$\left|x\right|_{S}\overset{\textrm{def}}{=}\prod_{p\in S}\left|x\...

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### Image of the ghost map of $p$-typical Witt vectors and $A$-ring structure of $W(A)$

For all ring with unit element $A$ let $W(A)$ be the ring of $p$-typical Witt vectors. Denote by $$\phi\;:\;W(A)\to A^{\mathbb{N}}$$
the ghost map, which is given by
$$\phi(a_0,a_1,a_2,\ldots)\;=\;(\...