# $$p$$-adic numbers

Let $$p$$ be a prime and let $$N_p=\{0,\ldots,p-1\}$$. Let $$\mathbb{N}$$ be the nonnegative integers.

Define the $$p$$-adic rationals $$\mathbb{Q}_p$$ to be the set of functions $$x:\mathbb{Z} \to N_p$$ for which there exists some $$k_x$$ such that $$x(k)=0$$ for all $$k<k_x$$. $$\mathbb{Q}_p$$ is a field for which addition and multiplication are defined in 1 and 2

For $$x \in \mathbb{Q}_p$$, define the $$p$$-adic valuation

$v_p(x)=\inf \{k \in \mathbb{Z}: x(k) \neq 0\}.$

This is a normalized discrete valuation: (see 3)

$$v_p(\mathbb{Q}_p \setminus \{0\}) = \mathbb{Z}$$,

If $$x(k)=0$$ for all $$k \in \mathbb{Z}$$, then $$v_p(x) = \infty$$.

Define the $$p$$-adic integers

$\mathbb{Z}_p = \{x \in \mathbb{Q}_p : v_p(x) \geq 0\}.$

Define the $$p$$-adic absolute value for $$x \in \mathbb{Q}_p$$ by

$\vert x \vert_p = p^{-v_p(x)}.$

If $$v_p(x)=\infty$$, then $$\vert x \vert_p=0$$.

The $$p$$-adic absolute value satisfies what is called an ultrametric/non-Archimedean property: for $$x,y \in \mathbb{Q}_p$$,

$|x+y|_p \leq \max\{|x|_p,|y|_p\}.$

For $$x \in \mathbb{Q}_p$$ and for $$k \in \mathbb{Z}$$, define $$p^k x \in \mathbb{Q}_p$$ by

$(p^k x)(j) = x(j-k). \qquad j \in \mathbb{Z}$

Then, for $$p$$-adic valuations,

$v_p(p^k x) = -k + v_p(x)$

and $$p$$-adic absolute values,

$|p^k x|_p = p^{-v_p(p^kx)} = p^{k-v_p(x)} = p^k |x|_p.$

# Topology

Expressed using the $$p$$-adic absolute value,

$\mathbb{Z}_p = \{x \in \mathbb{Q}_p: |x|_p \leq 1\}$

that is, the $$p$$-adic integers are a closed ball of radius 1 in the $$p$$-adic rationals.

Because $$v_p$$ is a discrete valuation, it follows that there is some $$\epsilon>0$$ such that

$\{x \in \mathbb{Q}_p: |x|_p \leq 1\} = \{x \in \mathbb{Q}_p: |x|_p < 1 + \epsilon\},$

which shows that $$\mathbb{Z}_p$$ is in fact an open set in $$\mathbb{Q}_p$$.

# Embedding $$\mathbb{Q} \hookrightarrow \mathbb{Q}_p$$

The additive group $$\mathbb{Q}_p$$ is a locally compact abelian group. $$\mathbb{Z}_p$$ is a compact set in $$\mathbb{Q}_p$$ so it has finite Haar measure, and it is an open set in $$\mathbb{Q}_p$$ so it has positive Haar measure. Therefore there is a unique unique Haar measure $$\mu_p$$ on $$\mathbb{Q}_p$$ such that $$\mu_p(\mathbb{Z}_p)=1$$.

$$\mathbb{Q}_p$$ is a field and $$\mathbb{Z}_p$$ is a ring. The multiplicative group of units of $$\mathbb{Z}_p$$ is

$\mathbb{Z}_p^\times = \{x \in \mathbb{Z}_p: |x|_p = 1\}.$

In particular,

$p^k \mathbb{Z}_p = \{x \in \mathbb{Q}_p: |x|_p \leq p^{-k}\}$

and

$p^k \mathbb{Z}_p^\times = \{x \in \mathbb{Q}_p: |x|_p = p^{-k}\},$

and one checks that

$\mathbb{Q}_p = \{0\} \cup \bigcup_{k \in \mathbb{Z}} p^k \mathbb{Z}_p^\times$

and

$\mathbb{Z}_p = \{0\} \cup \bigcup_{k \in \mathbb{N}} p^k \mathbb{Z}_p^\times.$

One checks

$\mu_p(p^k \mathbb{Z}_p)=p^{-k}$

and

$\mu_p(\mathbb{Z}_p^\times) = \frac{p-1}{p}.$

For a Borel set $$A$$ in $$\mathbb{Q}_p$$ and for $$x \in \mathbb{Q}_p$$,

$\mu_p(x \cdot A) = |x|_p \mu_p(A)$

For $$f \in L^1(\mathbb{Q}_p)$$ and $$x \in \mathbb{Q}_p \setminus \{0\}$$,

$\int_{\mathbb{Q}_p} f(x^{-1}y)d\mu_p(y) = |x|_p \int_{\mathbb{Q}_p} f(y) d\mu_p(y).$

# Multiplicative Haar measure

The multiplicative group

$\mathbb{Q}_p^\times = \{x \in \mathbb{Q}_p : x \neq 0\} = \mathbb{Q}_p \setminus \{0\}$

is a locally compact abelian group. The restriction of $$\mu_p$$ to the Borel $$\sigma$$-algebra of $$\mathbb{Q}_p^\times$$ is a Borel measure on $$\mathbb{Q}_p^\times$$.

One then proves (see 4) that

$d\nu_p(x) = \frac{p}{p-1} \dfrac{1}{|x|_p} d\mu_p(x)$

is a Haar measure on $$\mathbb{Q}_p^\times$$ with $$\nu_p(\mathbb{Z}_p^\times)=1$$.

Then it is proved that for $$s \in \mathbb{C}$$ with $$\mathrm{Re}(s)>-1$$,

$\int_{\mathbb{Z}_p^\times} |x|_p^s d\mu_p(x) = \dfrac{p-1}{p(1-p^{-1-s})}$

and that for $$\mathrm{Re}(s)>0$$,

$\int_{\mathbb{Z}_p^\times} |x|_p^s d\nu_p(x) = \dfrac{1}{1-p^{-s}}.$

# $$p$$-adic fractional parts

For $$x \in \mathbb{Q}_p$$, define $$[x]_p \in \mathbb{Q}_p$$ by

$[x]_p(k) = \begin{cases} x(k)&k<0\\ 0&k \geq 0 \end{cases}, \quad k \in \mathbb{Z},$

called the $$p$$-adic fractional part of $$x$$.

For $$x \in \mathbb{Q}$$, 5

$x - \sum_{p} [x]_p \in \mathbb{Z}.$

# $$p$$-adic Gaussians

References: 5 6 7