Questions tagged [p-adic]
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110 questions
0 votes
0 answers
52 views
Isometric map of affinoid p-adic algebras
Let $A = \mathbb{Q}_p\langle t_1, \dots, t_n \rangle = \mathbb{Q}_p\langle T_1, \dots, T_n \rangle/J$ be an $p$-adic affinoid algebra generated by $t_1, \dots, t_n$ with its norm being the quotient ...
- 1,607
7 votes
0 answers
188 views
$p$-adic distribution of $j$-invariants with formal CM
I'm trying to understand how "common" elliptic curves with potential formal complex multiplication (formal CM for short) are in the $p$-adic setting. Let me clarify the terminology: I say ...
- 411
4 votes
0 answers
179 views
Radius of convergence of solution to p-adic differential equation
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 $f(x) =...
- 41
3 votes
0 answers
179 views
Hecke compatibility for p-adic local langlands
Both in the global geometric Langlands as well as the local ($l \neq p$) Langlands formulated in the paper Geometrization of the local Langlands correspondence by Fargues and Scholze, there is a ...
- 131
10 votes
0 answers
267 views
Differences between Banach and analytic $p$-adic local Langlands correspondence
I have been trying to understand the categorical $p$-adic local Langlands paper [EGH 23] and I have a question regarding the differences between the Banach and analytic case of the conjectured ...
- 131
2 votes
1 answer
164 views
Frobenius antecedent of a differential module
Let $K$ be an ultrametric complete field of mixed characteristic, and let $F_{\rho}$ be the completion of $K(t)$ with respect to $\rho$-Gauss norm. After Christol and Dwork, for a differential module $...
4 votes
1 answer
222 views
Maximum modulus principle over the $p$-adic integers
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 \...
- 1,607
1 vote
1 answer
194 views
Convergence of a product in $\mathbb Q_2[[X]]$
I thought it would be very easy to prove, but in fact, I did not manage to prove or disprove this fact: the sequence of polynomials $$\left(\prod_{j=0}^k\big(1-2^{2^j}X\big)\right)_{k\in\mathbb N}$$ ...
- 4,286
1 vote
1 answer
190 views
Reduction of elliptic curves over local fields
Let $E$ be an elliptic curve defined over a p-adic local field $K$, with $j$-invarient $j(E)\in K$. Let $\mathscr{O}_K$ be the ring of integer of $K$. If $j(E)$ does not belong to $\mathscr{O}_K$, ...
- 95
4 votes
1 answer
240 views
Representations of $\mathrm{GL}_n(\mathcal{O})$ in functions on Grassmannians
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. The natural representation of the group $\GL_n(\...
- 23k
4 votes
1 answer
249 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 ...
- 23k
33 votes
3 answers
2k views
A funny metric over $\mathbb{N}$
$\DeclareMathOperator{\lcm}{lcm}$ Fiddling with numbers I realized that for positive integers $x$ and $y$, the quantity $$\Vert x,y \Vert=\frac{\lcm(x,y)}{\gcd(x,y)}$$ has these properties: $\Vert x,...
- 657
2 votes
0 answers
248 views
Computing the Dieudonné module of $\mu_p$ from Fontaine's Witt Covector
In Groupes $p$-divisibles sur les corps locaux, Fontaine introduced a uniform construction of Dieudonné modules through the definition of the Witt covector. Consider a perfect field $k$ of ...
- 389
1 vote
0 answers
165 views
Is the functor $\mathrm{Hom}(\mathrm{spec}\,k[x^{1/{p^\infty}}]/(x), -)$ from the category of finite commutative group schemes exact?
Question. Let $B \twoheadrightarrow C$ be a fully faithful homomorphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. ...
- 389
2 votes
3 answers
526 views
A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd
Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$ of cardinality $2m\ge 6$ where $m$ is odd. Question 1. Is it true that $G$ always has a subgroup $H$ of index 2 ...
- 15.2k
1 vote
1 answer
495 views
Unramified extension over $ \mathbb{Q}_{p} $
Let $\mathbb{Q}_{p}$ be a p-adic field such that $ p \neq 2 $. We knew that for every $ n=2m $ there exists exactly one unramified extension $ K $ of $ \mathbb{Q}_{p} $ of degree $ n $, obtained by ...
- 923
2 votes
0 answers
77 views
power of analytic function is still analytic in Krasner sense
In page 54 of his book, "Analytic elements in $p$-adic analysis" Escassut claims that if $f$ is analytic in Krasner sense in a set $D$ of a ultrametric field, so is $f^n$ for any positive ...
- 4,286
1 vote
0 answers
66 views
analytic continuation of Krassner's analytic element on punctured disk
Let $$ E=\{x\in\mathbb Q_p\mid |x|_p\le p^{-2}\}\setminus\left\{p^n\mid n\in\mathbb N,n\ge2\right\} $$ and $g$ be an analytic element (in Krassner's sense) of $$ F=\{x\in\mathbb Q_p\mid |x|_p<1\}\...
- 4,286
4 votes
1 answer
446 views
The points of $\operatorname{Spa}\mathbb{Z}_p$
$\DeclareMathOperator\Spa{Spa}$What are the points of $\Spa\mathbb{Z}_p$? I read in Scholze-Weinstein that this adic spectrum consists of 2 points, a special point, which corresponds to the pullback ...
- 3,346
2 votes
0 answers
198 views
Formulation of $p$-adic Haar measure decomposition
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\vol{vol}\DeclareMathOperator\diag{diag}$Suppose: $F$ is a non-archimedean local field, $\mathcal{O} \subset F$ its ring of integers, $\pi \in \mathcal{...
- 778
8 votes
0 answers
584 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 ...
- 778
2 votes
0 answers
169 views
Local Rankin-Selberg Zeta-function and Coates' p-adic L-Functions
$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\diag{diag}$ Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer. ...
- 778
5 votes
0 answers
306 views
A $p$-adic homotopy theory for non-simply connected spaces?
I'm looking to understand the state of the art for $p$-adic (unstable) homotopy theory of non-simply connected (non-nilpotent!) spaces. Ideally, I'd also like integral versions, e.g. things like ...
- 1,178
6 votes
0 answers
490 views
p-adic Hecke operators in the Iwahori-Hecke algebra $C_c(J\backslash G(F)/J)$
$\DeclareMathOperator\ch{ch}$Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer. I shall use $\kappa(F)$ to denote ...
- 778
1 vote
1 answer
125 views
Extension of morphisms in function fields
Let $k=\mathbb F_q\left(\left(\frac1T\right)\right)$, $\overline k$ be an algebraic closure of $k$ and $K$ be the completion of $\overline k$ for the $\frac1T$ valuation. Consider the morphism $\sigma:...
- 4,286
-2 votes
1 answer
815 views
$p$-adic number field $\mathbb{Q}_p $and algebraic numbers [closed]
As we all know, the complex number field $\mathbb{C}$ be a finite Galois extension field of the real number field that contains all algebraic numbers. I want to know the proof of the following ...
- 1
2 votes
0 answers
208 views
$G_K$-fixed points of sections of affinoids on the Fargues-Fontaine curve
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G_K=\mathrm{Gal}(\overline{K}/K)$ be its absolute Galois group. There are the Fargues-Fontaine analytic curves $Y_{FF}$ and $X_{FF}$ associated ...
- 663
19 votes
1 answer
2k views
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: ...
- 11.1k
5 votes
0 answers
484 views
A local model of a Shimura variety and a local Shimura variety
I have a question about the book on p-adic geometry by Scholze and Weinstein. There are two ‘local theories of Shimura varieties’ written in it. The one is a local model of a Shimura variety. This is ...
- 153
3 votes
1 answer
661 views
Algebraic numbers in all $\mathbb Q_p$ [duplicate]
Do there exist non-rational algebraic numbers that belong to $\mathbb Q_p$ for all prime $p$? If yes, can one characterize them? I spent several days for the first question, and I found nothing. The ...
- 4,286
5 votes
1 answer
550 views
Question about log and exp of a formal group law
Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $F$ be a Lubin–Tate formal group law defined over $K$ with endomorphism $f(T)$ corresponding to $\pi$ (a uniformizer of $K$). Then one can define ...
- 123
1 vote
0 answers
182 views
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 ...
- 123
2 votes
0 answers
392 views
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 =...
- 123
6 votes
1 answer
895 views
Vector bundles on the various sites of a preperfectoid
Let $X$ be a preperfectoid space over $\mathrm{Spa}(\mathbb{Q}_p,\mathbb{Z}_p)$. It has several associated sites, with successively finer topologies: $$X_{an} \subset X_{et} \subset X_{proet} \subset ...
- 663
1 vote
1 answer
402 views
Looking for an electronic copy of Huber's Bewertungsspektrum und rigide Geometrie
Lately I've been trying (and have failed) to find an electronic copy of Huber's Bewertungsspektrum und rigide Geometrie, which (from what I understand) is the original reference developing the basics ...
- 12.6k
1 vote
1 answer
501 views
Characters of p-adic units
Let $p$ be a prime and denote by $\mathbb{Z}_p^{\times}$ the group of $p$-adic units. Suppose that $\chi$ is a character $\chi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}^{\times}$. Then it is well ...
- 59
6 votes
1 answer
673 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)\;=\;(\...
- 333
1 vote
1 answer
210 views
Is the completion of the field generated by torsion points of a 1-dimensional formal group perfectoid?
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a 1-dimensional formal group defined over $\mathcal{O}_K$. Consider the field $K_\infty$ obtained by adjoining to $K$ all the solutions ...
- 663
3 votes
0 answers
145 views
Homology of a fiber as a cotorsion product
Let $K$ be a field. For any differentially graded coalgebra $A$ over $K$, any differentially graded right $A$-comodule $M$ over $K$ and any differentially graded left $A$-comodule $N$ over $K$ let $\...
- 1,721
6 votes
2 answers
1k 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 $X$? ...
- 663
4 votes
3 answers
250 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 ...
- 843
1 vote
0 answers
76 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 $P$-...
- 4,286
1 vote
0 answers
396 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 ...
- 433
3 votes
1 answer
156 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 ...
- 4,286
2 votes
0 answers
92 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 ...
- 4,286
6 votes
2 answers
905 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\...
- 4,286
3 votes
0 answers
132 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^...
- 4,286
3 votes
2 answers
830 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 $\...
- 749
3 votes
1 answer
594 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 ...
- 159
3 votes
0 answers
105 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 ...
- 507