$p$-adic
2進絶対値は、《2で何回割ることができるか》を表す数を $ n $ として
$$ \frac{1}{2 ^ n} $$
で定義される
# p-adic numbers
## Basic
### distance
function d : 2 elements $ ( x, y ) \rightarrow \mathbb{R} $
1. $ d(x,y) = 0 $ iff $ x = y $
2. $ d(x,y) = d(x,y) $
3. $ d(x,y) \le d(x,z) + d(z,y) $ $ \forall z \in X $
A set $ X $ together with metric d is called a $ metric\ space $ .
$$ ( X , d ) $$
### norm
1. $ || x || = 0 $ iff $ x = 0 $
2. $ || x * y || = || x || * || y || $
3. $ || x + y || \le || x || + || y || $
$$ d(x,y) = || x - y || $$
## Metrics
### Def. $ Order $
$$ {ord} _ p a \quad := \quad {greatest}\ m\quad |\quad a \equiv 0 \quad ( \mod p^m \ ) $$
### e.g.
$$ {ord} _ 5 250 = 3 , \quad {ord} _ 5 50 = 2 , \quad {ord} _ 5 36 = 0 $$
$$ \quad {ord} _ 2 24 = 3 , \quad {ord} _ 2 27 = 0 $$
### Def. $ p-adic\ norm $
$$ |x| _ p =
\begin{cases}
\frac{1}{p^{ {ord} _ p x}} \quad if x \neq 0; \\
0 \quad \quad if x = 0.
\end{cases}
$$
### e.g.
$$ |p| = p\ , \quad |p| _ p = \frac{1}{p} $$
### Def. $ non-Archimedean $
A norm is called $ \quad non-archimedean $
$$ ============================== (def) $$
$$ \forall x. \forall y. \quad || x + y || \le \max ( || x || , || y || ) $$
### Thm. $ Ostrowski $
$ \quad $
all norms on $ \mathbb{Q} $ are only two elements.
$ \quad $
$ |x| $ and $ |x| _ p $
### redefine p-adic norm
$ \quad $
$ \rho := \frac{1}{p} $
$ \quad $
$ | x | _ p = \rho ^ {ord _ p x} $
### $ trivial\ norm $
$$
\begin{cases}
|| 0 || = 0 \\
|| x || = 1 \quad if x \neq 0
\end{cases}
$$
### Thm. $ Ostrowski $
Every $ nontrivial $ norm $ || \ || $ on $ \mathbb{Q} $ $ \cong $ $ | \ \ | _ p $ for some prime p or p = $ \infty $
here,
$$ | \ \ | _ \infty \equiv | \ \ | $$
### Distance on Non-Archimedian
#### Triangle
**********************************************************************
* z *
* + *
* / \ *
* / \ *
* / \ *
* / \ *
* / \ *
* +-----------+ *
* x y *
* *
**********************************************************************
$ d(x,y) \le max(d(x,z),d(z,y)) $
let $ d(x,z) \gt d(z,y) $
so, $ \quad $ $ d(x,y) \le d(x,z) $
$ d(x,z) \le d(x,y) + d(y,z) \le max(d(x,y),d(y,z)) $
( $ \because $ Non-archimedian )
since $ d(x,z) \gt d(y,z) $ ,
so, $ \quad $ $ d(x,z) \le d(x,y) $
that is
$$ d(x,z) = d(x,y) $$
$$ ============================ $$
every triangle is an isosceles triangles
#### Circle
****************************************************************
* *
* .-. *
* | | *
* '-' *
* D(a,r-) *
* *
* *
****************************************************************
$ D(a,r ^ - ) := \{ x \in \mathbb{Q} \ | \ | x - a | _ p \le r \} $
$$ \forall b \in D(a,r ^ - ) . \quad D(a,r ^ - ) = D(b,r ^ - ) $$
$$ ========================== $$
all the inner point of D is the center of D
## Review of building up the complex numbers
p 進数 をコーシー列で完備化
## The field of $ p $ - adic numbers
# p-adic interpolation of the Riemann zeta-function
# Building up $ \Omega $
# p-adic power series
# Rationality of the zeta-function of a set of equations over a finite field