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.
245
questions
4
votes
1
answer
511
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 ...
- 6,834
3
votes
0
answers
81
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 ...
- 695
7
votes
0
answers
233
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. ...
- 111
4
votes
0
answers
184
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 ...
- 695
12
votes
1
answer
561
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 $1/(1-\alpha)$?
- 17.2k
1
vote
0
answers
88
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 ...
- 31
19
votes
1
answer
515
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: ...
- 9,480
9
votes
0
answers
419
views
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 ...
- 939
2
votes
0
answers
89
views
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 ...
- 573
8
votes
0
answers
147
views
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 ...
- 449
9
votes
0
answers
430
views
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'...
- 1,216
3
votes
0
answers
96
views
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 $...
- 303
3
votes
0
answers
152
views
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 ...
3
votes
0
answers
127
views
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 ...
- 1,201
1
vote
0
answers
69
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 ...
3
votes
0
answers
99
views
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
$$\delta_p(P) := \lim_{n \to \infty} \frac{|\{P(x) \bmod p^n : x = 1,\...
- 453
2
votes
0
answers
208
views
Existence of "nth root function" which is analytic
Let $K$ be a finite extension of $Q_p$. Let $q$ be the size of the residue field of $K$, and let $\pi$ be a uniformizer of $K$. Then $q/\pi$ is some power of $\pi$ up to a unit $u$ in $K$, say $q/\pi =...
4
votes
0
answers
176
views
Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?
When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...
- 2,389
3
votes
1
answer
314
views
Bernoulli distributions and $p$-adic measure on $K$
The $p$-adic field $\mathbb{Q}_p$ has topological basis of open sets of the form $a+p^N \mathbb{Z}_p$ for $0 \leq a \leq p^N-1$ and $N \in \mathbb{Z}$. These are indeed compact open sets. One can ...
- 754
0
votes
0
answers
105
views
A p adic limit of a binomial coefficient
Let $0 \leq a \leq p^n$ be a number coprime to p. Consider the following sequence of binomial coefficients:
$$B_k = \binom{p^{n+k}}{p^ka} $$
as $k\to \infty$. If I did the computation right, the p-...
- 6,834
4
votes
0
answers
164
views
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 ...
- 159
8
votes
1
answer
649
views
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 ...
- 6,672
2
votes
0
answers
70
views
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$. ...
- 821
2
votes
0
answers
76
views
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
...
- 2,997
3
votes
0
answers
124
views
$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}...
1
vote
0
answers
172
views
'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 ...
1
vote
0
answers
197
views
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$-...
2
votes
0
answers
203
views
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\...
- 1,216
4
votes
1
answer
415
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
$$\phi(a_0,a_1,a_2,\ldots)\;=\;(\...
- 155
5
votes
1
answer
354
views
Power of $2$ dividing a specialized Mittag-Leffler polynomial
While studying the so-called Mittag-Leffler Polynomials, denoted $M_n(x)$, I was looking into the sequence $\frac1{n!}M_n(n)$ which takes the following form
$$a_n:=\sum_{k=1}^n\binom{n-1}{k-1}\binom{n}...
- 37.8k
1
vote
0
answers
72
views
How to estimate the highest power of 2 in the partial sum of 2-adic $\log(-1)$ (i.e. $\sum_{i=1}^n\frac{2^i}{i}$)?
The estimate I wanna get is $$v_2(\sum_{i=1}^n\frac{2^i}{i})\geq\min_{t\geq n+1}\{t-v_2(t)\}\tag{*}$$
where $v_2$ is the 2-adic valuation, that is the highest power of 2 defined on $\mathbb{Q}$.
Set $$...
user178596
1
vote
1
answer
135
views
Is the completion of the field generated by torsion points of a 1-dimensional formal group perfectoid?
Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a 1-dimensional formal group defined over $\mathcal{O}_K$. Consider the field $K_\infty$ obtained by adjoining to $K$ all the solutions ...
- 643
3
votes
0
answers
247
views
A question on the Robba ring
Notation is as in the question:
https://math.stackexchange.com/questions/4090045/some-questions-about-the-robba-ring.
We define a new operator over the Robba ring as follows. Put $$c=\frac{pE(u)}{E(0)}...
- 66
12
votes
1
answer
373
views
Open immersion of affinoid adic spaces
If $R$ and $S$ are complete Huber rings with $\varphi: R \to S$ a continuous map, then is it true in general that if $\mathrm{Spa}(S, S^\circ) \to \mathrm{Spa}(R, R^\circ)$ is an open immersion of ...
- 377
1
vote
0
answers
133
views
A p-adic analogue of a result due to Kirillov
Let $k$ be a non-Archimedean local field with char$(k)=0$.
Let $N$ be the group of $k-$rational points of a unipotent algebraic group defined over $k$.
It is known that $N$ is a locally compact and ...
- 293
11
votes
1
answer
728
views
Where should I learn about the p-adic L-functions of elliptic curves?
Where is the best place to learn about the p-adic L-functions of Elliptic Curves? Doing a bit of research I have found books like "An Introduction to Cyclotomic Fields" by Washington, but ...
- 2,187
2
votes
0
answers
57
views
Name for a logarithmic ratio of roots
I'm trying to find a name for the following quantity that came up in my research. I've asked some people and looked around myself but can't find a name, yet it seems like something that has probably ...
- 109
3
votes
1
answer
129
views
A sequence modified from the Catalan recurrence and $2$-adic valuations
Earlier, I posted this MO question to which Max Alekseyev found counter-examples. I realized that there was some error in my computational programming with Maple. Oh, well $\dots$ I have scaled down ...
- 37.8k
3
votes
1
answer
185
views
Tweaking the Catalan recurrence and $2$-adic valuations
Among many descriptions of the Catalan numbers $C_n$, let's use the recursive format $C_0=1$ and
$$C_{n+1}=\sum_{i=0}^nC_iC_{n-i}.$$
Then, the $2$-adic valuation of $C_n$ is computed by $\nu_2(C_n)=s(...
- 37.8k
22
votes
1
answer
2k
views
Condensed criterion for sheafiness of adic spaces
Multiple times in talks about condensed mathematics (e.g. the Masterclass talks, Clausen's RAMpAGe talk), it is stated that the derived structure sheaf given by the condensed formalism "fixes&...
- 323
2
votes
2
answers
278
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 ...
- 2,135
9
votes
1
answer
874
views
On the definition of the etale site of an adic space
I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces".
First ...
- 228
6
votes
1
answer
410
views
What is good $t$-adic like topology on $\mathbb{C}(t)$?
Each function $f\in\mathbb{C}(t)$ can be rewritten in the form $f = a_{k}t^{k}+\ldots+a_0+a_1t+\ldots$, $k\in\mathbb{Z}$ and it is possible to define the topology with the open prebase at zero
$V_{n,v,...
- 303
1
vote
0
answers
69
views
How to analyze the roots of two variable $p$-adic power series $f(x,y)$ in the maximal ideal $m$?
Let $K \subseteq \mathbb{Q}_p$ be the $p$-adic field, $O_K$ be the ring of integer and $m$ be the maximal ideal.
Consider the single variable $p$-adic power series $f(x)=\sum_{n=0}^{\infty}a_n x^n \...
- 754
4
votes
0
answers
204
views
How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series?
How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series ?
We know that if $f(x)=\sum_{i=0}^{\infty} a_ix^i$ be a power series over $p$-adic field, then the Newton ...
- 754
3
votes
1
answer
71
views
Are injective analytic maps between non-archimedean spaces open?
Let $\Omega$ be a non-archimedean complete field, $n\in\mathbb N$ and $f:\Omega^n\to\Omega^n$ be an injective analytic map.
Is the application $f$ open?
In the complex case, this is a consequence of a ...
- 2,997
1
vote
0
answers
85
views
Doubt regarding invariance of discrete absolute value under automorphism
I have been reviewing some basic algebraic number theory and $p$-adic analysis, and the following thought crossed my mind: if $F/ \mathbb Q$ be a finite Galois extension, and $\eta$ be a $\mathbb Q$-...
- 679
1
vote
0
answers
118
views
I'm looking for a proof of Polya-Bertrandias Theorem
I'm looking for a proof of Polya-Bertrandias rationality criterion in english (not the one from Amice).
- 11
0
votes
0
answers
66
views
Are analytic functions in several variables open mappings?
It is well known that an analytic function $f$ defined on an open set $\Omega$ is an open mapping: for every open subset $U$ of $\Omega$, $f(U)$ is an open set of $\mathbb C$. Is this result still ...
- 2,997
2
votes
0
answers
148
views
What is the p-adic Plancherel measure?
What I know as the Plancherel measure for a group is a measure on the spectrum of $G$, aka the set of irreducible representations - at least for finite groups, this makes perfect sense.
Now, this ...
- 6,834