Topological construction of p-adic integers

By ellipticcurves

One of the interesting things to observe is what multiplication by $p$ does to $p$ -adic integers:

$p(u_1, u_2, u_3, \ldots) = (p u_1, p u_2, p u_3, \ldots).$

But what matters is the congruence class of the components modulo $p^r,$ and $pu_1 \equiv 0 \pmod p,$ $pu_{r}\equiv pu_{r-1} \pmod{p^r},$ so

$p(u_1, u_2, u_3, \ldots) = (0, p u_1, p u_2, \ldots).$

Iterating gives us

$p^n (u_1, u_2, u_3, \ldots) = (\underbrace{0, \ldots, 0}_{n\text{ times}}, p^n u_1, p^n u_2, \ldots).$

This is, of course, a lot easier when using the series notation.

So, we can tell that $p^n$ divides $u$ precisely when the first $n$ components of $u$ are zero. As a corollary, we obtain that if $u\neq 0$ , then there exist a largest $n$ such that $p^n$ divides $u$ (i.e., $p^{n+1}$ does not divide $u$ ). This $n$ is precisely the number of the initial components of $u$ that are congruent to zero in their respective ${\mathbb Z}/p^r{\mathbb Z}.$

Definition. The number $n$ as above is called the $p$ -adic valuation of $u,$ and is denoted by $\mathrm{ord}_p(u)$ :

$\mathrm{ord}_p(u) := \mathrm{max}\{n\in {\mathbb N} | p^n\text{ divides }u\text{ in }{\mathbb Z}_p \}.$

We set $\mathrm{ord}_p(0) = \infty.$

The $p$ -adic valuation function satisfies the following properties:

$\mathrm{ord}_p(u) = \infty$ if and only if $u=0,$
$\mathrm{ord}_p(uv) = \mathrm{ord}_p(u)\mathrm{ord}_p(v),$ and
$\mathrm{ord}_p(u+v) \geq \mathrm{min}(\mathrm{ord}_p(u), \mathrm{ord}_p(v)).$

The last property is referred to as the ultrametric property for reasons to be explained right now.

Given any number $0 < \sigma < 1$ (customarily, $\sigma = 1/p$ ), we can define

$|u|_p := \sigma^{\mathrm{ord}_p(u)}$ .

Then the properties of $\mathrm{ord}_p$ become

$|u|_p \geq 0$ for all $u$ , with $|u|_p = 0$ if and only if $u=0,$

$|uv|_p = |u|_p\, |v|_p,$ and

$|u+v|_p \leq \mathrm{max}(|u|_p, |v|_p).$

In other words, $|\bullet|_p: {\mathbb Z}_p \to {\mathbb R}$ is an ultrametric norm. We can also define a metric

$d_p(u,v) := |u-v|_p,$

which makes ${\mathbb Z}_p$ into an utrametric space. The topology (convergence of sequences) induced by this norm does not change if we choose a different $\sigma$ , so from now on, $\sigma = 1/p$ . Then

$|u|_p = \frac{1}{p^{\mathrm{ord}_p(u)}}.$

In other words, the higher power of $p$ divides you, the smaller you are.

Note that this norm can be restricted to the rational integers ${\mathbb Z}$ contained in ${\mathbb Z}_p$ . The ring of integers is not complete under this norm, and therefore we can consider its completion, i.e., the smallest complete normed space containing ${\mathbb Z}$ . Then the standard construction of the completion coincides for the $p$ -adic norm with the construction of $p$ -adic integers as sequences. Specifically, the condition that $u_r \equiv u_{r+s} \pmod{p^r}$ translates into the Cauchy criterion $d_p(u_r, u_{r+s}) \to 0$ as $r\to 0$ for sequences.

In light of this norm, the series notation also starts to make sense. Indeed, the truncated series form a Cauchy sequence, since $b_r p^r + b_{r+1} p^{r+1} + \cdots + b_{r+s} p^{r+s}$ has norm no greater than $1/p^r$ , which goes to zero as $r$ goes off to infinity. Thus the infinite series we work with converge with respect to the ultrametric $p$ -adic norm.

Tags: p-adic

This entry was posted on December 26, 2008 at 3:49 am and is filed under p-adic numbers. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Elliptic Curves

Topological construction of p-adic integers

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