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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4 votes
1 answer
149 views

Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operators" by Coleman and Iovita

Let $G$ be an abelian variety with good reduction over a finite extension $K$ of $\mathbb{Q}_p$. In equation (2.4) on page 179 of my edition of "The Frobenius and monodromy operators for curves ...
19 votes
2 answers
1k views

P-adic C* algebras

I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-...
0 votes
1 answer
351 views

Does $\sum_{n \geq 0} a_n x^n=\sum_{n \geq 0} b_nx^n$ imply $a_n=b_n$ for vector-tuple power series?

My reference is Infinite series in p-adic fields by Keith Conrad. Corollary 5.6. If $f(x)=\sum_{n≥0} a_nx^n$ has a positive radius of convergence in the $p$-adic field $\mathbb Q_p$ then $f$ is ...
2 votes
1 answer
157 views

Integral over the space of $p$-adic matrices

$\DeclareMathOperator\Mat{Mat}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers with a uniformizer $\pi$. Let $|\cdot|\colon \mathbb{F}\to \mathbb{R}$ be ...
4 votes
0 answers
107 views

Projective reduction of image of power series is algebraic?

Let $K$ be a non-archimedean field with closed unit disk $\mathcal{O}\subset K$, open unit disk $\mathfrak{m}\subset \mathcal{O}$ and residue field $k = \mathcal{O}/\mathfrak{m}$. Examples to keep in ...
4 votes
0 answers
160 views

Bezout-type theorem for $p$-adic analytic plane curves

Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
3 votes
1 answer
157 views

Approximating $p$-adic power series by polynomials

Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...
2 votes
0 answers
111 views

p-adic Banach space and complete tensor product

Let $p$ be a prime and $\mathbb{C}_{p}$ the completion of the algebraic closure of the $p$-adic number field $\mathbb{Q}_p$. Let $M$ be a $\mathbb{Q}_p$-Banach space. We denote by $M\mathbin{\widehat{\...
4 votes
1 answer
122 views

Partition of unity for analytic manifolds over non-Archimedean local fields

I am looking for a reference to the following fact which, I hope, is correct. Let $X$ be a compact analytic manifold over a non-Archimedean local field. Let $X=\cup_\alpha U_\alpha$ be a finite open ...
1 vote
0 answers
76 views

The bound for zeros of the composition of polynomials and analytic functions

Suppose $K$ is a number field, and $A\in M_n(K)$. $v$ is a place of $K$, and $f_1,\cdots,f_n$ are analytic functions (one variable) on $m_v\mathcal O_{K,v}$, satisfying: $\frac{\mathrm d \bf {f}}{\...
2 votes
1 answer
144 views

How to get a ball in the nonvanishing locus of a polynomial in $\mathbb Z_p[x_1,\cdots,x_n]$ canonically?

Suppose $f\in \mathbb Z_p[x_1,\cdots,x_n]$, and consider $D(f):=\{(𝑥_1,…,𝑥_𝑛)∈ℤ^𝑛_𝑝:𝑓(𝑥_1,…,𝑥_𝑛)≠0\}\subset \mathbb Z_p^n$. How to calculate a radius $r$ from the datum of $f$ such that $D(f)$...
0 votes
1 answer
221 views

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 ...
3 votes
0 answers
172 views

Wondering if Monsky-Washnitzer ever published a result claimed to be forthcoming in a later paper

At the very end of the paper Formal Cohomology I by Monsky and Washnitzer, they write the following: "In some sense, the operator $\psi$ applied to a power series gives it "better growth ...
6 votes
1 answer
461 views

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)...
4 votes
0 answers
159 views

Notion of connected components for $\mathbb{Q}_p$-points of algebraic variety

Is there an interesting notion of connected components for the $\mathbb{Q}_p$-points of an algebraic variety over $\mathbb{Q}_p$? By "interesting" I mean a notion satisfying the following. ...
2 votes
0 answers
132 views

$p$-adic Banach group algebra

Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
1 vote
1 answer
262 views

Formal series which are always zero

Let $(k, |\cdot|)$ be a complete field with a non-Archimedean norm, not necessarily algebraically closed. Define the Tate algebra as follows: \begin{align*} k \langle T_1, \dots, T_n \rangle = \{ \...
0 votes
0 answers
67 views

Space of non-archimedean characters is nonempty

Let $k$ be an algebraically closed complete non-archimedean field. Let $\mathcal{O}_k$ be its ring of integers. Suppose that $A$ is a $k$-Banach algebra, and $B$ is its closed unitary ball. Note that $...
62 votes
11 answers
13k views

Elementary results with p-adic numbers

I'm giving a talk for the seminar of the PhD students of my math departement. I actually work on Berkovich spaces and arithmetic geometry but, of course, I cannot really talk about that to an audience ...
2 votes
1 answer
239 views

Twist of the Tate Curve

Suppose we have an elliptic curve $E$ over $K$, an $l$-adic field. Say that $|j(E)|>1$ where $|.|$ is the $l$-adic valuation. By the theory of the Tate curve $E$ is isomorphic over $L$ to a Tate ...
10 votes
1 answer
602 views

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 ...
2 votes
1 answer
157 views

Smooth function on $\mathbb{Z}_{p}$

Let $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ be a function such that $\delta_{a}f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ is a locally constant function for any $a\in\mathbb{Z}_{p}$, where $\delta_{a}f(x):...
1 vote
0 answers
98 views

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 ...
2 votes
0 answers
51 views

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(...
3 votes
1 answer
393 views

$\lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p$ 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 ...
1 vote
1 answer
138 views

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 $\mathbb Q$. ...
17 votes
4 answers
10k views

What is the p-adic valuation of a number?

There seem to be two conflicting definitions for p-adic valuation in the literature. Firstly, for any non-zero integer n, we have $\nu=\nu_p(n)$ is the greatest non-negative integer such that $p^\nu$ ...
2 votes
2 answers
467 views

A p-adic logarithm as a limit of discrete logs

I've been searching for something similar to the argument below for about a week now and I just must be missing out on the right key words. Can someone point me in the right direction and/or let me ...
1 vote
1 answer
51 views

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 ...
3 votes
0 answers
238 views

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 ...
2 votes
1 answer
289 views

$p$-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 ...
1 vote
1 answer
89 views

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 ...
3 votes
1 answer
435 views

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 ...
13 votes
2 answers
712 views

Lindemann theorem for Artin-Hasse exponential

Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
0 votes
0 answers
160 views

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 $f(x)=\sum_{n>0}a_nx^n \in \mathbb Z_p[[x]]$ ...
2 votes
0 answers
205 views

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 ...
5 votes
1 answer
569 views

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)\;=\;(\...
2 votes
1 answer
129 views

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 ...
8 votes
0 answers
340 views

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 ...
5 votes
1 answer
346 views

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 ...
55 votes
14 answers
19k views

'Important' applications of p-adic numbers outside of algebra (and number theory).

Surely, $\mathbb{Z}_p$ and $\mathbb{Q}_p$ (and their extensions) are very important for algebra and number theory. Do they have any important applications outside of algebra (that I could easily ...
2 votes
0 answers
174 views

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 ...
0 votes
0 answers
170 views

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} ...
4 votes
1 answer
578 views

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 ...
22 votes
4 answers
4k views

Special values of $p$-adic $L$-functions.

This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes. My understanding is that nowadays there are conjectures which essentially ...
3 votes
0 answers
129 views

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 ...
7 votes
0 answers
283 views

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. ...
4 votes
0 answers
257 views

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 ...
12 votes
1 answer
592 views

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 $1/(1-\alpha)$?
1 vote
0 answers
127 views

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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