All Questions
22 questions
3
votes
1
answer
180
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 ...
5
votes
0
answers
197
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 ...
2
votes
1
answer
150
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 $D(f)$...
- 785
3
votes
1
answer
437
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 ...
- 1,451
2
votes
1
answer
356
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 ...
- 91
12
votes
3
answers
2k
views
What is the value of $p$-adic $\zeta$-function at positive integer point?
$p$-adic zeta function is a $p$-adic interpolation of the Riemann $\zeta$-function for the values $\zeta(1−k)$, $k\ge 1$ (see $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz) ...
- 12.3k
1
vote
0
answers
94
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 $$...
user178596
5
votes
0
answers
328
views
Analytic continuation of $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the p-adics
Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$.
What can be said about an analytic continuation "in the form of Mittag-...
- 461
2
votes
2
answers
2k
views
A multidimensional version of Hensel's lemma? (for more than one polynomial)
The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}_p[x]$ and $a \in \mathbb{Z}_p$ satisfy
$$
|f(a)|_p < | f'(a) |_p^2.
$$
Then there is a unique $\alpha \in \mathbb{Z}_p$...
- 3,625
4
votes
2
answers
2k
views
Automorphisms of $\mathbb C_p$
I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.
If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that ...
- 3,998
4
votes
0
answers
124
views
Finite dimensional irreps of $p$-adic groups
What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$?
One knows such a representations cannot be smooth, so probably the examples will be ...
- 81
11
votes
0
answers
1k
views
Nick Katz observation: "the rationality of the zeta function!"
In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-...
- 3,107
3
votes
1
answer
2k
views
What is $p$-adic Fourier series?
Q1: Can we define Fourier series for a function $\mathbb{Z}_p\to \mathbb{Q}_p$?
Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion:
$$B_n(\{x\})=-\frac{...
- 12.3k
11
votes
1
answer
774
views
2-adic Logarithm and Resistance of n-dimensional Cube
Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is
$$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=0}^{n-1}\frac1{{n-1\choose k}}.$$
The ...
- 12.3k
9
votes
0
answers
408
views
Transcendence of the $p$-adic number $\sum_{n\ge0}a^{2^n}$
Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.
Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb ...
- 3,998
8
votes
0
answers
895
views
Lemma in Scholze-Weinstein
In the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7:
Lemma: Let $K$ be a ...
- 89
4
votes
1
answer
1k
views
Iwasawa logarithm and analytic continuation
I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$.
...
- 539
2
votes
0
answers
166
views
Relative Leopoldt defect
Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$.
Is there a bound of the Leopoldt defect of $M$ ...
- 1,066
3
votes
0
answers
176
views
Carlitz factorials and Euler-like series
Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+...
- 3,998
3
votes
1
answer
499
views
Trivial p-adic measures
I am looking at p-adic distributions, and in this case p-adic measures. To say that $\mu$ is a distribution means that the arguments of $\mu$ are compact open subsets of $\mathbb{Z}_p$, $\mu$ is ...
- 43
4
votes
0
answers
174
views
Simultaneously using the real and 2adic norms
In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to ...
user13113
0
votes
0
answers
121
views
Computing a projection of a $p$-adic plane curve
Given a prime $p$ and a polynomial equation $f(x,y)=0$ with rational coefficients, I would like to obtain a precise description of the set of all numbers $y\in\mathbb Q_p$ such that the equation has a ...
- 1,021