Skip to main content
MathOverflow

Questions tagged [p-adic-analysis]

Ask Question

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

315 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
2 votes
0 answers
183 views

Let $K=\mathbb{Q}_3(t)$ be the finite extension of the $3$-adic number field $\mathbb{Q}_3$, where $t=3^{1/13}$. I want to know if the polynomial $$f(x)=(x^9-t^2)^3-3^4x+3^3t^5 \in K[x]$$ is ...
Learner's user avatar
  • 450
3 votes
0 answers
83 views

Let $p$ be an odd prime and $\mathbb{Z}_{p}$ be the ring of $p$-adic integers. Let $\mathbb{F}_{p}$ be the finite field of order $p$ and let $\mathbb{F}_{p}[[T]]$ be the ring of formal power series ...
stupid boy's user avatar
  • 917
2 votes
0 answers
147 views

I'm following Scholze-Weinstein's Berkeley notes (https://www.math.uni-bonn.de/people/scholze/Berkeley.pdf). And there is a theorem by Huber (Thm 2.3.3) in the notes that says the adic spectrum $\...
wxhan's user avatar
  • 21
1 vote
0 answers
48 views

I encountered the following question when studying the cohomology of a char p variety, which is really outside of my area of expertise. Notation: $\mathbb{F}=\mathbb{F}_p$ bar, $W$ its Witt ring, $K=W[...
RFJ's user avatar
  • 21
5 votes
0 answers
249 views

Let $K$ be a field complete with respect to a non-Archimedean absolute value $|\cdot|$. To develop analysis in $K$, we need the notion of a disk in $K$. There is nothing mysterious at first glance: ...
Yuri Bilu's user avatar
  • 1,354
14 votes
1 answer
480 views

A nonarchimedean valued field $K$ is said to be spherically complete if, for any nested sequence $B_1 \supseteq B_2 \supseteq \dots$ of balls in $K$, the intersection $\bigcap_{i = 1}^\infty B_i$ is ...
CJ Dowd's user avatar
  • 283
2 votes
0 answers
81 views

Let $X$ be an arbitrary set. Let $H = c_0(X, \mathbb{Q}_p)$ be the $p$-adic Banach space with sup norm. Let $\langle \cdot, \cdot \rangle$ be a symmetric, nondegenerate $\mathbb{Q}_p$-bilinear form on ...
Luiz Felipe Garcia's user avatar
  • 1,607
4 votes
0 answers
184 views

I am working on a problem that seems to reduce to determining (to certain precision) the radius of convergence of a particular solution of a p-adic differential equation. In particular, we have $f(x) =...
Piotr's user avatar
  • 41
12 votes
1 answer
310 views

Let $p$ be a prime number, and let $H(\mathbb{C}_p)$ denote the ring of power series $f(T)\in \mathbb{C}_p[[T]]$ such that $f(T)$ converges in an open ball of radius $1$ about $0$. n.b. that this is ...
Snacc's user avatar
  • 423
1 vote
0 answers
111 views

As context, I'll start with summarizing and simplifying the section of "UMAC: Fast and Secure Message Authentication", by Black et al.(https://www.cs.ucdavis.edu/~rogaway/papers/umac-full....
Jim Apple's user avatar
  • 111
2 votes
0 answers
99 views

Let $G=GL_n(F)$ over a p-adic field $F$, use $\mathcal{S}(G)$ to denote the compactly supported locally constant functions on $G$. Let $\Omega_n:=N_nA_n\omega_n N_n$ be the open dense Bruhat cell of $...
user550518's user avatar
  • 341
4 votes
1 answer
281 views

Let $k$ be a perfect field of characteristic $p$. Let $X$ be a quasi-projective smooth variety over the Witt ring $W=W(k)$ ($K=\mathrm{Frac}(W)$). Let $\mathcal X$ be the $p$-adic formal completion of ...
Yang's user avatar
  • 471
2 votes
0 answers
174 views

Let $F$ be a $d$-dimensional formal group of finite height over the ring of $p$-adic integers $\mathbb{Z}_p$. Let $\mathfrak{m}_{\mathbb{C}_p}$ be the unit disk in $\mathbb{C}_p$. Since $F$ the formal ...
Learner's user avatar
  • 450
7 votes
0 answers
205 views

In Dwork's paper "Bessel functions as $p$-adic functions of the argument", a certain constant $\gamma$ arises as a matrix entry in his calculations of a Frobenius structure that he ...
Daniel Pomerleano's user avatar
  • 4,046
7 votes
1 answer
455 views

Let $\mathbb{Z}_p$ denote the $p$-adic integers. Does $\operatorname{GL}_2(\mathbb Z_p)$ have an open solvable subgroup $H $? By solvable I mean that there exist a chain of subgroups $$1=G_0\lhd G_1 \...
jacm125's user avatar
  • 83
1 vote
1 answer
208 views

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{tr}$Let $A\in \SL(2,\mathbb{Z})$. Inside $\SL(2,\mathbb{Z}_p)$, we can consider profinite powers $A^r$, where $r\in\widehat{\mathbb{Z}}$. For ...
stupid_question_bot's user avatar
  • 7,785
7 votes
0 answers
331 views

I've been reading the proof of Ax–Sen–Tate from Caruso - An introduction to p-adic period rings; it says that $\mathbb{C}_p^G = \mathbb{Q}_p$. The proof contains subtle computations which I want to ...
Andy's user avatar
  • 631
1 vote
1 answer
134 views

Let $C$ be the completion of an algebraic closure of $\mathbb{Q}_p$, and let $\mathcal{O}_C$ be its ring of integral elements. Let $A$ be an $\mathcal{O}_C$-algebra that is torsion and almost ...
dgulotta's user avatar
  • 953
0 votes
1 answer
235 views

Paterson-Stockmeyer algorithm If we need to compute a high-degree polynomial expression, such as: $$ P(y) = \sum_{k=0}^{B} a_k y^k $$ the Paterson-Stockmeyer algorithm can process the powers in ...
AC.PR's user avatar
  • 27
2 votes
1 answer
116 views

In Gouvea book $p$-adic numbers, on can find this corollary (7.4.11) Let $f(X) = 1+a_1X+a_2X^2+a_3X^3+\cdots$ be a power series which converges on the closed ball of radius $c = p^m$. Let $m_1, m_2, \...
joaopa's user avatar
  • 4,306
1 vote
1 answer
303 views

Are there a polynomials $f_1,...f_n \in \mathbb{Z}_p[x_1,...x_n]$ with there coeficients $p$-adic integers s.t. A map $F:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p^n$ defined by $f_1,...f_n$ satisfy the ...
George's user avatar
  • 378
3 votes
1 answer
332 views

My question is how to prove the affine $n$-space over $p$-adic number $\mathbb{Q}_p$ is simply connected. To be precise, Let $X$ be $p$-adically analytic manifold, $f:X\rightarrow \mathbb{A}^n_{\...
George's user avatar
  • 378
9 votes
1 answer
307 views

Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity $w$ if for every homomorphism $ f:...
stupid boy's user avatar
  • 917
2 votes
1 answer
295 views

Let $(x)_0=1$ and $(x)_n=x(x-1)\cdots(x-n+1)$ for $n=1,2,3,\ldots$. The signed Stirling numbers of the first kind, $s(n,k)$ with $n\ge k\ge0$, are defined by $$(x)_n=\sum_{k=0}^ns(n,k)x^k.$$ Question. ...
Zhi-Wei Sun's user avatar
  • 18.1k
1 vote
0 answers
132 views

Let $p$ be a prime. Let $K$ be a number field and $L/K$ be an infinite extension which is not necessarily Galois. Suppose that the automorphism group $\text{Aut}(L/K)$ of $L/K$ is $p$-adic analytic, i....
stupid boy's user avatar
  • 917
2 votes
1 answer
172 views

Let $G$ be $p$-adic group and let $G \rightarrow GL(V)$ be a representation. For example, $V$ is a quadratic $\mathbb{Q}_p$-space and $G$ is the associated orthogonal group. Take a point $v \in V$, ...
Sentem's user avatar
  • 71
11 votes
1 answer
477 views

Disclaimer: my knowledge of $p$-adic analysis/geometry is minimal. Consider a smooth, complete curve $C$ of genus $g$ over $\mathbb{C}_{p}$, denote by $J$ its Jacobian and consider the embedding $C\...
Giulio Bresciani's user avatar
  • 1,000
3 votes
0 answers
93 views

Let $\Bbb Q_p$ be a p-adic field and let any element $x$ of $\Bbb Q_p$ be associated with a unique element of $\Bbb Z_p$ via the quotient / equivalence relation $\forall n\in\Bbb Z:p^nx\sim x$ Then in ...
Robert Frost's user avatar
  • 426
0 votes
0 answers
117 views

Let $k$ be a field with characteristic $0$. For every set $X$, let $\mathcal{B}(X)$ be the set of (possibly infinite) matrices $T = (T_{x,y})_{x,y \in X}$ with coefficients in $k$ such that in each ...
Luiz Felipe Garcia's user avatar
  • 1,607
4 votes
1 answer
228 views

Consider $\mathbb{Z}_p$ the $p$-adic integers. Let $f\in\mathbb{Z}_p[x]$ be an arbitrary polynomial in one variable. Write $f(x) = \sum_{k}a_kx^k$. Is it true that $\|f\|:= \max_k |a_k|_p = \sup_{t \...
Luiz Felipe Garcia's user avatar
  • 1,607
0 votes
0 answers
117 views

Given a non increasing function $\psi$ the $\psi$ approximable points in $\mathbb{R}^n$ is defined as $W(\psi)=\{x\in\mathbb{R}^n:|qx-p|<\psi(q)\}$ for infinitely many $(q,p)\in \mathbb{Z}^m\times\...
User5's user avatar
  • 11
4 votes
1 answer
421 views

Is it true that any separated quasi-compact rigid-analytic variety embeds into a proper one? For my purpose, the base field is a $p$-adic number field. I have seen Huber's universal compactification ...
SmileLee's user avatar
  • 101
3 votes
2 answers
333 views

Let $K$ be a complete non-archimedean field. A norm on a finite dimensional vector space $V$ is a function $| \cdot | : V \to \mathbf{R}$ which satisfies the usual norm properties (with the non-...
Thiago's user avatar
  • 273
4 votes
1 answer
311 views

I'm reading IRREDUCIBLE COMPONENTS OF RIGID SPACES (by Conrad). In this paper he defines the irreducible component of a rigid variety $X$ to be reduced image of a connected component of $\tilde X$ (...
Richard's user avatar
  • 1,051
2 votes
1 answer
237 views

$\DeclareMathOperator\Mat{Mat}$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 ...
asv's user avatar
  • 23.1k
4 votes
0 answers
140 views

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 ...
Jef's user avatar
  • 1,068
5 votes
0 answers
238 views

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 ...
Ashvin Swaminathan's user avatar
  • 757
3 votes
1 answer
221 views

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 ...
Ashvin Swaminathan's user avatar
  • 757
2 votes
0 answers
215 views

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{\...
user521844's user avatar
  • 21
1 vote
1 answer
501 views

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 ...
MAS's user avatar
  • 928
4 votes
1 answer
255 views

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 ...
asv's user avatar
  • 23.1k
1 vote
0 answers
98 views

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}}{\...
Richard's user avatar
  • 1,051
2 votes
1 answer
197 views

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 $D(f)$...
Richard's user avatar
  • 1,051
4 votes
1 answer
242 views

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 ...
Vik78's user avatar
  • 1,135
3 votes
0 answers
225 views

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 ...
Vik78's user avatar
  • 1,135
4 votes
0 answers
267 views

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. ...
Jacques's user avatar
  • 605
2 votes
0 answers
165 views

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 ...
Luiz Felipe Garcia's user avatar
  • 1,607
1 vote
1 answer
308 views

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 = \{ \...
Luiz Felipe Garcia's user avatar
  • 1,607
0 votes
0 answers
98 views

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 $...
Luiz Felipe Garcia's user avatar
  • 1,607
2 votes
1 answer
295 views

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 ...
did's user avatar
  • 667

1
2 3 4 5
…
7