# Questions tagged [p-adic-number-theory]

In mathematics the $p$-adic number system for any prime number $p$ extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.

1,225 questions
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### What is the meaning of $\rho \nu_{\rho}$?

I am reading the paper. On page 14, section 3.1, I am trying to understand the notation $\rho \nu_{\rho}$, where $\rho$ is a supercuspidal representation $GL_n(F)$, $F$ is a non-archimedean local ...
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### In what sense is the L2 norm the canonical norm on $\mathbb{R}^n$ but the max-norm is the canonical norm on $\mathbb{Q}_p^n$?

I understand that the L2 norm is natural to consider on $\mathbb{R}^n$ both because of its geometric intuition, and because it is induced by the dot product on $\mathbb{R}^n$. I also understand that a ...
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### chain of inequalities with $p$-adic anaysis

I'm reading a paper of recurrences sequences. In the Lemma 2 there is a chain of inequalities, which i've locked in the blue box in the image, which I do not understand how they are justified. If ...
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### Making the subgroup of units of a topological ring a topological group

I'm a beginner in the theory of topological algebra (I'm reading something about it in Robert's "A course in $p$- adic analysis"). At page 24, the author states that if $A$ is a topological ring, the ...
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### $\Bbb{Z}_px\subseteq\overline{\Bbb{Z}x}$

I'm studying $p$-adic integers and in the proof of the fact that closed subgroups of the additive group $\Bbb{Z}_p$ are ideals (see Robert's "A course in $p$-adic analysis", pp.23) I've found the ...
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Here is a problem i came across. Prove that $$\sum_{i=1}^{n}\frac{1}{i}$$ is not an integer for $n \geq 2$. The book olympiad number theory by Justin Stevens says that after writing $\frac{1}{i}$ as ...
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### Additive characters of $\mathbb{C}_p$

Consider $x \in \mathbb{C}_p$ with $|x|<1$ then for $a \in \mathbb{Z}_p$ we have the characters $$a \mapsto (1+x)^a$$ where $(1+x)^a= \exp(a\log_p(1+x))$ My question is : it's possible to ...
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### Lubin-Tate formal groups are $p$-divisible groups

I am trying to understand how to see whether a given formal group is $p$-divisible. Let $A$ be a complete noetherian local ring with maximal ideal $\mathfrak{m}$ and residue field $k$ of ...
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### Looking for this article by Paulo Ribenboim

While going through a forum post (http://mathforum.org/kb/message.jspa?messageID=40112), I found the following paper mentioned: 87a:12014 12J10 13A18 Ribenboim, Paulo (3-QEN) Equivalent forms of ...
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### Equality with $p$-adic analysis

I'm reading a paper of recurrences sequences and I could not understand why the equality mentioned in the underlined lines in the image. I think that it has to do with $p$-adic analysis. If someone ...
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### Norms on fields

I'm doing an introductory module in number theory, and came across the definition of a norm on a field. It seems to agree with the definition of a norm on a vector space over a field (just view the ...
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### Set $X(\mathbb{Q}_p)$ of $\mathbb{Q}_p$-valued Points not Empty

My question refers to a step in AriyanJavanpeykar's argumentation in following former thread of mine: https://mathoverflow.net/questions/325014/irreducible-smooth-proper-one-dimensional-schemes-...
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### On the Newton polygon for Laurent series

I'm stuck with an understanding of what should be the Newton polygon for a Laurent series. I'm reading ''An introduction to G-function" by Dwork and he dedicates only three pages to Newton polygons ...
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### About Bernoulli polynomials

My question is about Bernoulli numbers and Bernoulli polynomials in the $p$-adic context. In general in fact Bernoulli numbers are defined as global object so they do not depend on $p$. If $B_k(x)$ is ...
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### About continuous functions on $p$-adic fields

Consider $K/ \mathbb{Q}_p$ a finite extension of the field of $p$-adic numbers. If for every such an extension $f_K: K \to K$ is continuous can we extend these functions to $\mathbb{C}_p$? My idea ...
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### Whether the p-adic power series $\sum \frac{(n!)^n}{1+(n!)^{2n}} x^n$ converges p-adically

p-adic convergence: The p-adic power series $\sum \frac{1}{n!} x^n$ is divergent. But what about the p-adic power series $\sum \frac{(n!)^n}{1+(n!)^{2n}} x^n$? Does it converge p-adically? Answer: ...
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### Understanding the $p$-part of the discriminant of a totally real number field with a single prime above $p$

Let $K$ be a totally real Galois number field, and suppose there is only one prime above $p$, with ramification index $\leq p-1$. If $K_p$ is the completion of $K$ at the prime above $p$, the claim ...
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### primitive representation of integers over $\mathbb{Z}_p$

In Cassel's book "Rational quadratic forms" page 235 he claims that the form $$x_1^2 + x_2^2 +5(x_3^2 + x_4^2)$$ primitively represents $3 \cdot 2^{2m}$ over $\mathbb{Z}_p$ for every prime $p$ and ...
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### Embedding $\mathbb C_p$ into $\mathbb C$ and vice versa…?

I'm reading Washington's book on cyclotomic fields, and he mentions that it is sometimes convenient to embed $\mathbb{C}_p$ into $\mathbb C$ and vice versa. In my mind, $\mathbb C_p$ and $\mathbb C$ ...
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### Reference request: $p$-adic unit for $p≠2$ is a square in $\mathbb{Q}_p \iff$ its first digit is a quadratic residue modulo $p$.

I'm looking for a book to reference that contains the statement in the title. Many thanks for your help.
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### Torsion-free abelian groups, tensor product and $p$-adic integers

I'm studying torsion-free abelian groups and I know (see Fuchs, "Infinite Abelian Groups", vol. $2$, pp $154$) that, if $\mathbb{Z}_p$ is the set of $p$- adic integers and $\mathbb{Z}_{(p)}$ denotes ...
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