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Non-archimedean absolute values and completions

Absolute Values

We start by defining absolute values on $\mathbb Q$, but much of this extends naturally to absolute values on other fields. An absolute value on $\mathbb Q$ is a function $\mathbb Q \rightarrow [0,\infty)$ such that:

  1. positive definiteness: $|x| = 0$ iff $x = 0$,
  2. multiplicativity: $|x y| = |x| |y|$ for all $x, y \in \mathbb Q$
  3. triangle inequality: $|x + y| \leq |x| + |y|$ for all $x,y \in \mathbb Q$.

If, in place of (3), $|\cdot|$ satisfies the strong triangle inequality: $|x + y| \leq \max\{|x|, |y|\}$, then we say $| \cdot |$ is a non-archimedean absolute value.

Example

Let $p$ be a prime, and define the standard $p$-adic absolute on $\mathbb Q$ by
$$
\left|p^{\ell} \frac{a}b\right| = p^{-\ell}.
$$
Here we assume that $a$ and $b$ are rational integers not divisible by $p$. $| \cdot|_p$ is a non-archimedean absolute value. Note that if $n$ is a rational integer, then $|n| \leq 1$.

Completions

Given an absolute value $|\cdot|$ on $\mathbb Q$, we say a sequence $(x_n)$ is Cauchy if given $\epsilon > 0$, there exists $N \in \mathbb N$ so that $n, m > N$ implies $|x_n – x_m| < \epsilon$. Similarly, we say that $(x_n)$ converges to zero if given $\epsilon > 0$ there exists $N \in \mathbb N$ so that $n > N$ implies $|x_n| < \epsilon$. Using the standard operations on sequences, the set of Cauchy sequences $\mathcal C$ is a (commutative) ring, and the set of sequences which converge to zero, $\mathcal Z$ is a maximal ideal. It follows that $\mathcal C / \mathcal Z$ is a field, and we call this the completion of $\mathbb Q$ with respect to the absolute value $| \cdot |$. We may embed $\mathbb Q$ into $\mathcal C / \mathcal Z$ by the diagonal embedding $x \mapsto (x,x,\cdots)$. The fact that $\mathbb Q$ is dense in $\mathcal C / \mathcal Z$ allows us to extend $| \cdot |$ to a unique absolute value on $\mathcal C / \mathcal Z$.

The completion of $\mathbb Q$ with respect to the usual absolute value is $\mathbb R$.
The completion of $\mathbb Q$ with respect to $| \cdot |_p$ is called the field of $p$-adic numbers and denoted $\mathbb Q_p$.

The Topology of $\mathbb Q_p$

A basic neighborhood in $\mathbb Q_p$ is a set of the form $B(x, \epsilon) = \{y \in \mathbb Q_p : |y – x|_p \leq \epsilon\}$. Since $| \cdot |_p$ takes values in the discrete set $\{p^{-n} : n \in \mathbb Z\}$, every basic neighborhood is both open and closed, since for instance
$$
\{y : |y|_p \leq p^{-n} \} = \{y : |y|_p < 2p^{-n} \}.
$$

Let’s investigate the ball $B(1, p^{-1})$ by looking at what integers it contains. Clearly $1 \in B(1, p^{-1})$. The condition $|y – 1| \leq p^{-1}$ is equivalent to $y \equiv 1 \bmod p$. Similarly, $B(1, p^{-n})$ contains all the integers $y$ such that $y \equiv 1 \bmod p^n$. For instance, when $p = 7$, the integers $1$ and $344$ are close, a distance of $7^{-3}$ from each other, since $344 – 1 = 343 = 7^3$.

For a slightly more complicated ball, consider $B(1/2, p^{-2})$. This is by definition
$$
\{y \in \mathbb Q_p : |y – 1/2|_p \leq p^{-2} \}.
$$
From the definition of $| \cdot |_p$, if $y$ is a rational number so that $|y – 1/2|_p = p^2$, then there exist integers $a$ and $b$, both relatively prime to $p$, so that $y – 1/2 = p^2 a/b$. More generally, if $|y – 1/2|_p \leq p^2$ then there exist integers $a$ and $b$, with $b$ relatively prime to $p$, so that $y – 1/2 = p^2 a/b$. It follows that the rational numbers in $B(1/2, p^{-2})$ are of the form $1/2 + p^2 r$ where $r$ is a rational number which contains no factors of $p$ in the denominator when written in lowest terms.

The General Setup

Suppose $K$ is a number field of degree $d$ over $\mathbb Q$ and suppose $\mf O$ is the ring of integers of $K$. Given a prime ideal $\mf p \subseteq \mf O$ we can define an absolute value $| \cdot |_{\mf p}$ by fixing $q > 0$ and setting
\[
|x|_{\mf p} = q^{-n} \qquad \mbox{where} \qquad x \in \mf p^n \setminus \mf p^{n+1}.
\]
The completion of $K$ with respect to $| \cdot |_{\mf p}$ is denoted $K_{\mf p}$.

The ring of integers in $K_{\mf p}$ is defined to be $\mf o_{\mf p} = \{x \in K_{\mf p} : |x|_{\mf p} \leq 1\}$. The ideal $\mf m_{\mf p} \subseteq \mf o_{\mf p}$ given by $\mf m_{\mf p} = \{x \in \mf o_{\mf p} : |x| < 1 \}$. The units in $\mf o_{\mf p}$ are given by $U_{\mf p} := \{x \in \mf o_{\mf p} : |x|_{\mf p} = 1 \}$.

The field $\mf o_{\mf p} / \mf m_{\mf p}$ is a finite field of size $p^f$ for some rational prime $p$ and positive integer $f.$ It is convenient to take $q$ in the definition of $| \cdot |_{\mf p}$ to be $q := p^f.$

Often times, when we are working with a fixed non-archimedean absolute value, we drop the subscript $\mf p$ from various quantities. When in this situation we assume $(K, |\cdot|)$ is a field complete with respect to $|\cdot|$ and with integers $\mf o$, maximal ideal $\mf m$ and units $U$. Unless specifically indicated otherwise we take $q = [\mf o : \mf m]$. The maximal ideal is principal, and we set $\varpi$ to be a generator or uniformizer for the ideal: $\mf m = \varpi \mf o$. Clearly then $|\varpi| = 1/q$.

$p$-adic electrostatics

Suppose $k$ is a local field with ring of integers $\mathfrak o$. Let $\mf m \subseteq \mf o$ denote the unique maximal ideal, and define $q = [\mf o: \mf m]$.  The maximal ideal is principal and we fix a generator, or uniformizer, $\varpi$ so that $\mf m = \varpi \mf o$. There is a unique absolute value $| \cdot |$ on $k$ with
$$
| \varpi | = \frac{1}{q}.
$$
In this context $\mf o = \{ \alpha : |\alpha| \leq 1\}$ and $\mf m = \{\alpha : |\alpha| < 1\}$. As a locally compact abelian group, $k$ has a Haar measure $\mu$ which can be made unique by specifying $\mu(\mf o) = 1$.

Imagine two charged particles, $p$-adic electrons if you will, identified with $\alpha, \alpha’ \in \mf o$ and whose interaction energy is given by
$$
E(\alpha, \alpha’) = -\log|\alpha’ – \alpha|.
$$
This vaguely corresponds with our intuition as to how electrons ($p$-adic or not) should behave in the sense that the energy is minimized when the distance between $\alpha$ and $\alpha’$ is maximal, and the energy is infinite if the electrons are on top of each other, that is when $\alpha = \alpha’$. Since we have restricted $\alpha$ and $\alpha’$ to $\mf o$, the farthest apart they can be is $| \alpha – \alpha’| = 1$, and in this case the interaction energy is 0. This observation is central in understanding how these particles behave when we introduce thermal fluctuations to the mix.

Moving from $2$ particles to multiple particles, there are three main systems or ensembles we will consider.

The Microcanonical Ensemble
The system contains $N$ particles at a specified energy $E$.
The Canonical Ensemble
The system contains $N$ particles at a specified temperature $T$. Energy can be exchanged with a heat bath, and is now variable.
The Grand Canonical Ensemble
The system is now in contact with a heat bath and a particle reservoir so that $E$ and $N$ are both variable. The temperature $T$ and a new quantity, the chemical potential, are fixed and control the average energy and particle number.

The Microcanonical Ensemble

In this setting, the energy of the $N$ particles located at the coordinates of $\boldsymbol \alpha \in \mf o^N$ is given by
$$
E(\bs \alpha) = -\sum_{m < n} \log|\alpha_n - \alpha_m|. $$ All states with the same energy are assumed to be equally probable, and the main problem for the micro canonical ensemble is the determination of the measure of the set with prescribed energy. For reasons that will become apparent, it is sometimes useful to deal with the exponentiated energy $$ e^{-E(\bs \alpha)} = \prod_{n