Questions tagged [p-adic]

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6
votes
2answers
741 views

Vector bundles on adic spaces

Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? ...
3
votes
3answers
188 views

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

Let $K/F$ be a finite extension of local fields (of characteristic 0). Is it true that the quotient group $K^\times/ F^\times$ is always compact? I understand that if the extension is cyclic, it is ...
1
vote
0answers
57 views

Continous morphisms of a local field with conditions in positive characteristic

Let $P$ be a an irreducible polynomial of $k:=\mathbb F_q(T)$, $\Omega_P$ be the completion of an algebraic closure $\overline{k_P}$ of $k_P$, the completion of $k$ for the topology induced by the $P$-...
1
vote
0answers
283 views

On the paper “Patching and the p-adic local Langlands correspondence”

$\DeclareMathOperator\GL{GL}$The question is about the paper: Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin - Patching and the $p$-adic local Langlands correspondence. Let $F$ be a finite ...
3
votes
1answer
113 views

Analytic p-adic functions that take an algebraic value

Suppose it exists $r\in\mathbb R$ such that the non constant p-adic function $f(z)=\sum_{n\ge0}a_nz^n$ ($a_n\in\mathbb C_p$) is defined on $\mathcal D=\{z\in\mathbb C_p\mid v_p(z)>r\}$. Does it ...
2
votes
0answers
58 views

Zeroes of the Euler series

Consider a prime $p$. Let $f$ be the Euler series defined by $f(z)=\sum_{n\ge0}n!z^n\}$. It is defined and analytic over $\mathcal D=\{z\in\mathbb C_p\mid v_p(z)>-\frac1{p-1}\}$. I try to check if ...
6
votes
2answers
566 views

Zero of the exponential p-adic

Consider the $p$-adic exponential defined over $\mathbb C_p$. One knows $\exp$ is analytic in the domain $\mathcal D=\{z\in\mathbb C_P\mid v_p(z)>\frac1{p-1}\}$. Does it exist an element $z_0\in\...
3
votes
0answers
104 views

Composition in function fields

Let $k=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. One has the map: $\circ:k\times\{v\in k\mid\deg(v)>0\}\to k$ defined by $f\circ g=\sum_{n\ge-m}a_ng^{-n}$ where $f=\sum_{n\ge-m}a_n\frac1{T^...
3
votes
2answers
415 views

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

This question is inspired from the post linked below: Can an algebraic number on the unit circle have a conjugate with absolute value different from 1? What I am curious about is the following: let $\...
1
vote
1answer
184 views

Integrality certification for product of two matrices $A B^{-1}$

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...
3
votes
0answers
71 views

Freeness of completed homology over universal deformation ring

In Theorem 7.4 of the paper "patching and the $p$-adic Langlands program for $\mathrm{GL}2(\mathbf{Q}_p)$" (arXiv link), it is proved that in the minimally ramified case, the completed homology of ...
1
vote
0answers
70 views

irrationality of Bessel function in $p$-adic

Let $J_0(z)=\sum_{n\ge 0}\frac{(-1)^n}{n!^2}\left(\frac z2\right)^{2n}$ be the Bessel function considered in $\mathbb C_p$. Let $\alpha\in\mathbb Q^*$ be in the convegence disk of $J_0$. Is $J_0(\...
0
votes
0answers
105 views

continuous isomorphism of $p$-adic field

In Cassels-Frolich, one can read this theorem (page 57): Let $K$ be a finite finite separable extension of the valued field $(k,v)$. Let $\overline k$ be the completion of $k$ and $(K_j)_{1\le j\le r}$...
0
votes
0answers
51 views

Comparison of growth of entire functions in a $p$-adic field

Let $p$ be a prime number, $\Omega_p$ be the spherically complete extension of $\mathbb C_p$. Consider two entire functions on $\Omega_p$: $f(z)=\sum_{n\ge 0}a_nz^n$ and $g(z)=\sum_{n\ge0}b_nz^n$. ...
2
votes
0answers
183 views

A complete Tate Huber ring is Banachizable (maybe not)?

I have questions of technical nature. A complete Tate Huber ring is a complete topological (commutative) ring $A$ admitting an open subring $A_0$ whose topology is the $\varpi A_0$-adic topology, for ...
2
votes
1answer
118 views

Modulus of growth in $p$-adic spherically complete field of $\mathbb C_p$

Let $F$ be the spherically complete extension of $\mathbb C_p$ and $(a_n)_{n\in\mathbb N}$ be a sequence of $\mathbb C_p$ such that for all $r\in\mathbb R$, one has $$\lim_{n\to+\infty}|a_n|_pr^n=0.$$ ...
7
votes
1answer
321 views

P-adic functions on annuli

It is known that a complex analytic function defined on an annulus, say, takes its maximum on the boundary. Does an analogue hold for $p$-adic analytic functions? More precisely suppose we have a ...
1
vote
0answers
168 views

Extension of $\mathbb C_p$?

Let $K/\mathbb Q$ be a finite algebraic extension. Consider a prime $p$ and $v$ be a valuation of $K$ above the valuation $v_p$ of $\mathbb Q_p$. Denote by $K_v$ a completion for the valuation $v$ of $...
2
votes
0answers
65 views

Extension of valuations and completion

Let $L/K$ be a finite extension of fields where $K$ is a number field. Let $v$ be a valuation of $K$ and denotes by $K_v$ its completion for this valuation. One denotes $\tilde v$ be the valuation $...
3
votes
0answers
126 views

Mixed characteristic in symplectic geometry

Are there any mixed-characteristic phenomena in symplectic geometry/mirror symmetry? There are papers on symplectic geometry by Abouzaid (inspired by Kontsevich--Soibelman, I believe) in which there ...
1
vote
0answers
147 views

Automorphisms of $\mathbb C_p$ with constraints

In Automorphisms of $\mathbb C_p$, K. Conrad showed that there exist uncountably many $\mathbb Q_p$-automorphisms of $\mathbb C_p$. I have a quite similar question. Let $(a_n)_{n\in\mathbb N}$ and $(...
3
votes
1answer
285 views

Analytic continuation of a $p$-adic function

Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be sequences of $\mathbb Q_p$ such that the function $f:z\in\mathbb Q_p\to\sum_{n\ge0}a_nz^n+b_nz^{n+1}$converges in $\{|z|_p<1\}$. Assume ...
7
votes
1answer
675 views

Convergence of a $p$-adic series

Let $K$ be a local field of characteristic $0$ with valuation $v$. I think $$\lim_{\substack{q\in K\\q\to1}}\sum_{n\ge0}\prod_{j=1}^n\frac{q^j-1}{q-1}$$ converges to $\sum_{n\ge0}n!\in K$ but I did ...
2
votes
0answers
63 views

Invariant compact in division ring

Let $K$ be a discrete valued (with discrete valuation $v$) complete local division ring with ring of valuation $V$. Let $F$ be a compact subset of $V$. Suppose that for all $x\in F$ and for all $y\in ...
11
votes
1answer
755 views

Is the p-adic Lindemann-Weierstrass Conjecture still open?

The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$ be distinct $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\...
17
votes
1answer
416 views

Centraliser of an absolute Galois group

Let $K$ be a finite extension of $\mathbb{Q}_p$. Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ trivial ? If yes, how can I ...
5
votes
1answer
295 views

p-adic L-functions for (dual of) fine Selmer Groups

If $R(E/\Bbb Q_{\infty})$ is the fine Selmer group and $Y(E/\Bbb Q_{\infty})$ is its dual, then we know that $Y(E/\Bbb Q_{\infty})$ is a finitely generated $\Lambda$-module and by a theorem of Kato, ...
7
votes
1answer
293 views

How to deduce Bernstein-Zelevinsky classification from the Langlands one

I am trying to understand the Langlands classification. To that end, I am trying to find how I could deduce the Bernstein-Zelevinsky classifcation from the second description of the Langlands ...
5
votes
0answers
140 views

Arithmetic of Cuspidal Reps. Fundamental non split stratum and simple stratum

I started to read Colin Bushnell's notes on this title. The last theorem in the 3rd section claims that if $\pi$ is an irr. smooth rep. of $GL_N(F)$ containing a non-split fundamental stratum, then $\...
2
votes
1answer
1k views

History of Fargues-Fontaine curve

In this paper, Pierre Colmez wrote about some history of the Fargue-Fontaine curve. In this schedule of London Number Theory Study Group, Fargues was said to give a talk on November 15th on " Where ...
1
vote
0answers
63 views

Automorphisms in function fields with $\mathbb F_q[T]$ globally invariant

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q((1/T))$ for the valuation $-\deg$. Does there exist uncountably many automorphisms of $\Omega$ that let $\mathbb F_q[T]$ stable ...
2
votes
0answers
207 views

Crystalline cohomology over arbitrary perfect ground fields

In preparing a set of notes, I realize there is a simple question for which I cannot find an easy answer; likely due to simple ignorance. Any introductory text on etale cohomology notes that ...
5
votes
1answer
150 views

$p$-adic valuation of $P_n(p)$ if $P_n$ is the $n$th Legendre polynomial

In 3-adic valuation of a sum involving binomial coefficients the $p$-adic valuation of $P_n(p)$ has been obtained. Computations indicate that $M_n(p)=\sum\binom{n}{k}^2(p-1)^k$ for odd $p$ has the ...
16
votes
2answers
512 views

3-adic valuation of a sum involving binomial coefficients

Let $$a(n) = \sum_{0 \leq k \leq n} {n \choose k}{{n+k} \choose k},$$ and define $b(n) = \nu_3 \bigl(a(n)\bigr)$, where $\nu_3$ is the $3$-adic valuation. About twenty years ago or so, I discovered (...
4
votes
0answers
342 views

Cyclotomic Extension of a Perfectoid Space

Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more ...
1
vote
1answer
239 views

Specialisation of Z_p[[S_1,…,S_n,X_1,…,X_d]]

Let $R \colon= {\Bbb Z}_p[[S_1,...,S_n,X_1,...,X_d]]$ be the $(n+d)$-variable formal power series ring over ${\Bbb Z}_p$. Choose an arbitrary ideal ${\frak b}$ of $R$ such that ${\mathrm{ht}}({\...
8
votes
0answers
293 views

$p$-adic representations of the fundamental group of a smooth proper curve over a finite field

This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations $$ \pi_1(C)\...
1
vote
1answer
147 views

null infinite product in the p-adic setting

Let $p$ be a prime, $\mathbb C_p$ be the completion of a algebraic closure of $\mathbb Q_p$ and $(u_n)_{n\in\mathbb N}$ be a sequence of $\mathbb C_p$ converging towards $0$. Suppose that for all $n\...
6
votes
0answers
324 views

What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
1
vote
0answers
155 views

Kitagawa's p-adic modular symbols for different weights: a confusing observation

References are to K. Kitagawa, "On standard $p$-adic $L$-functions of families of elliptic cusp forms", Contemp. Math. 165, 1994. Let $\mathcal O$ be the ring of integers in a finite extension of $\...
7
votes
0answers
382 views

$p$-adic uniformisation of abelian varieties

In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement: Let $A$ over $\mathbf{Q}_p$ be an abelian variety ...
5
votes
0answers
737 views

Equivalent definitions of the Hasse invariant

As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant. Let me start by recalling one definition: Let $E\to S$ be an elliptic curve in ...
5
votes
2answers
711 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 $\mathbb{Q}...
2
votes
0answers
224 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 $\...
16
votes
2answers
1k views

Witt-vector vectors

I've never really made my way in any detail through the Witt-vector construction. I did read all the articles that a quick Google and MSN search turned up, and none seemed to address it, but I could ...
1
vote
0answers
85 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 ...
0
votes
0answers
124 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 ...
0
votes
0answers
193 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 ...
1
vote
1answer
637 views

Why is every l-adic Galois representation conjugate to one over the l-adic integers? [closed]

Why is every l-adic Galois representation $$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Q}_{l})$$ conjugate to one over the l-adic integers? $$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$
12
votes
0answers
541 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 ...