\(p\)-adic Numbers¶

Summary

There is not just one way to complete the rational numbers – besides the real numbers \(\mathbb{R}\), for every prime \(p\) there are the \(p\)-adic numbers \(\mathbb{Q}_p\). In this world, "close to zero" means "divisible by high powers of \(p\)".

Prerequisites¶

Rings and Fields

1. A Different Metric¶

How do we measure the "distance" between two rational numbers? The usual answer uses the ordinary absolute value \(|x|\): the number \(1/1000\) is close to \(0\) because \(|1/1000|\) is small.

But there is a completely different way to define "closeness" – one based on divisibility rather than size.

Definition. Let \(p\) be a prime. The \(p\)-adic valuation \(v_p(n)\) of a non-zero integer \(n\) is the highest power of \(p\) dividing \(n\):

\[ v_p(n) = \max\{k \geq 0 \mid p^k \mid n\} \]

For example, with \(p = 3\): \(v_3(54) = 3\) (since \(54 = 2 \cdot 3^3\)), \(v_3(7) = 0\), \(v_3(81) = 4\).

The \(p\)-adic absolute value is then:

\[ |x|_p = p^{-v_p(x)} \quad \text{for } x \neq 0, \qquad |0|_p = 0 \]

The astonishing feature: the more powers of \(p\) a number contains, the smaller its \(p\)-adic absolute value becomes. For \(p = 5\):

Number \(x\)	\(v_5(x)\)	\(\\|x\\|_5\)	\(\\|x\\|\) (ordinary)
\(1\)	\(0\)	\(1\)	\(1\)
\(5\)	\(1\)	\(1/5\)	\(5\)
\(25\)	\(2\)	\(1/25\)	\(25\)
\(625\)	\(4\)	\(1/625\)	\(625\)
\(1/5\)	\(-1\)	\(5\)	\(0.2\)

Intuition is turned on its head: \(625\) is \(5\)-adically tiny (close to \(0\)), but ordinarily large!

2. Convergence Turned Upside Down¶

The \(p\)-adic absolute value defines a metric on \(\mathbb{Q}\): \(d_p(x, y) = |x - y|_p\). In this metric, convergence statements hold that seem absurd from the perspective of analysis.

Example: The series \(\sum_{n=0}^{\infty} p^n\) converges in \(\mathbb{Q}_p\).

The partial sums \(S_N = 1 + p + p^2 + \cdots + p^N\) converge \(p\)-adically because the summands \(p^n\) tend to \(0\) \(p\)-adically for large \(n\): \(|p^n|_p = p^{-n} \to 0\). In fact, the series converges to:

\[ \sum_{n=0}^{\infty} p^n = \frac{1}{1-p} = \frac{-1}{p-1} \]

So in \(\mathbb{Q}_5\), \(1 + 5 + 25 + 125 + \cdots = -\frac{1}{4}\). This sounds paradoxical, but is \(5\)-adically perfectly correct.

The Ultrametric Inequality¶

The \(p\)-adic absolute value satisfies a property stronger than the triangle inequality:

\[ |x + y|_p \leq \max(|x|_p, |y|_p) \]

This is the ultrametric inequality (or "strong triangle inequality"). It has startling consequences:

Every triangle in \(\mathbb{Q}_p\) is isosceles (the two longest sides have equal length)
Every point of a "disc" is its centre
Series converge if and only if their terms tend to \(0\) (no ratio test needed!)

3. Construction of \(\mathbb{Q}_p\)¶

The real numbers \(\mathbb{R}\) arise as the completion of \(\mathbb{Q}\) with respect to the ordinary absolute value \(|\cdot|\): one adds the limits of all Cauchy sequences.

Entirely analogously: the \(p\)-adic numbers \(\mathbb{Q}_p\) are the completion of \(\mathbb{Q}\) with respect to the \(p\)-adic absolute value \(|\cdot|_p\).

Formally: \(\mathbb{Q}_p\) is the quotient field of Cauchy sequences in \((\mathbb{Q}, |\cdot|_p)\) modulo null sequences. Every element of \(\mathbb{Q}_p\) can be uniquely written as a \(p\)-adic expansion:

\[ x = \sum_{n=k}^{\infty} a_n p^n \quad \text{with } a_n \in \{0, 1, \ldots, p-1\}, \quad k \in \mathbb{Z} \]

This is like a "decimal expansion", but infinite to the left rather than to the right. The ordinary decimal representation \(0.333\ldots = 1/3\) has finitely many digits before the decimal point and infinitely many after. In \(\mathbb{Q}_p\) it is the other way round: finitely many digits after the "decimal point" (negative powers of \(p\)) and possibly infinitely many before.

4. Ostrowski's Theorem¶

How many essentially different absolute values are there on \(\mathbb{Q}\)?

Theorem (Ostrowski, 1916). Every non-trivial absolute value on \(\mathbb{Q}\) is equivalent to one of the following:

The ordinary absolute value \(|\cdot|\) (whose completion yields \(\mathbb{R}\))
A \(p\)-adic absolute value \(|\cdot|_p\) for some prime \(p\) (whose completion yields \(\mathbb{Q}_p\))

That is: \(\mathbb{R}\) and the \(\mathbb{Q}_p\) (for all primes \(p\)) are the only completions of \(\mathbb{Q}\). Every place – every way to "enlarge" \(\mathbb{Q}\) – corresponds either to the Archimedean absolute value (\(\infty\)-place) or to a \(p\)-adic absolute value (\(p\)-place).

Hasse's Principle

In number theory one often writes \(v \in \{\infty, 2, 3, 5, 7, \ldots\}\) for the places of \(\mathbb{Q}\). The local fields \(\mathbb{Q}_v\) (with \(\mathbb{Q}_\infty = \mathbb{R}\)) together form a complete picture of the rational numbers – one sees \(\mathbb{Q}\) "from all sides simultaneously".

5. The \(p\)-adic Integers \(\mathbb{Z}_p\)¶

The valuation ring of \(\mathbb{Q}_p\) consists of all elements with \(p\)-adic absolute value \(\leq 1\):

\[ \mathbb{Z}_p = \{x \in \mathbb{Q}_p \mid |x|_p \leq 1\} = \left\{ \sum_{n=0}^{\infty} a_n p^n \mid a_n \in \{0, \ldots, p-1\} \right\} \]

\(\mathbb{Z}_p\) is a local ring with the unique maximal ideal \((p) = p\mathbb{Z}_p\). The residue field is:

\[ \mathbb{Z}_p / p\mathbb{Z}_p \cong \mathbb{F}_p \]

The \(p\)-adic integers have an alternative description as a projective limit:

\[ \mathbb{Z}_p = \varprojlim_n \mathbb{Z}/p^n\mathbb{Z} \]

An element of \(\mathbb{Z}_p\) is thus a compatible system \((a_1, a_2, a_3, \ldots)\) of residue classes: \(a_n \in \mathbb{Z}/p^n\mathbb{Z}\) with \(a_{n+1} \equiv a_n \pmod{p^n}\).

Properties of \(\mathbb{Z}_p\):

\(\mathbb{Z}_p\) is a principal ideal domain (in fact a discrete valuation ring)
The units are \(\mathbb{Z}_p^\times = \{x \in \mathbb{Z}_p \mid |x|_p = 1\} = \mathbb{Z}_p \setminus p\mathbb{Z}_p\)
\(\mathbb{Z} \subset \mathbb{Z}_p\) is dense (every \(p\)-adic integer is the limit of a sequence of integers)
\(\mathbb{Z}_p\) is compact (as a topological space)

6. Hensel's Lemma¶

One of the most powerful tools of \(p\)-adic analysis is Hensel's lemma – the \(p\)-adic version of Newton's method.

Theorem (Hensel). Let \(f \in \mathbb{Z}_p[x]\) be a polynomial. If \(a \in \mathbb{Z}\) is a simple root of \(f\) modulo \(p\) (i.e., \(f(a) \equiv 0 \pmod{p}\) and \(f'(a) \not\equiv 0 \pmod{p}\)), then there is a unique root \(\alpha \in \mathbb{Z}_p\) of \(f\) with \(\alpha \equiv a \pmod{p}\).

The idea: from an approximate solution modulo \(p\), an exact solution in \(\mathbb{Z}_p\) is constructed step by step – by iteratively "lifting" modulo \(p^2\), \(p^3\), \(p^4\), and so on.

Example: Does \(\sqrt{2}\) exist in \(\mathbb{Q}_7\)? We check: \(3^2 = 9 \equiv 2 \pmod{7}\) and \(2 \cdot 3 = 6 \not\equiv 0 \pmod{7}\). By Hensel's lemma, \(\sqrt{2} \in \mathbb{Z}_7\) exists. By contrast, \(x^2 \equiv 2 \pmod{5}\) has no solution, so \(\sqrt{2} \notin \mathbb{Q}_5\).

Hensel as a lever

Hensel's lemma reduces many \(p\)-adic questions to finite computations modulo \(p\). Instead of working in the infinite field \(\mathbb{Q}_p\), it often suffices to compute in the finite field \(\mathbb{F}_p\) – and then lift.

7. The Local-Global Principle¶

The central idea: information about \(\mathbb{Q}\) can be obtained from the combination of all local information (over \(\mathbb{R}\) and all \(\mathbb{Q}_p\)).

Hasse–Minkowski theorem. A quadratic form over \(\mathbb{Q}\) has a non-trivial solution in \(\mathbb{Q}\) if and only if it has a solution in \(\mathbb{R}\) and in \(\mathbb{Q}_p\) for all primes \(p\).

This local-global principle works perfectly for quadratic forms – but not always. For cubic equations and elliptic curves, it can fail: there are curves that have points everywhere locally but no global rational point. The measure of this failure is the Tate–Shafarevich group \(\Sha\) – a deep object of arithmetic geometry.

Local Conditions in Wiles' Proof¶

In Wiles' proof, the local fields \(\mathbb{Q}_p\) play a fundamental role:

Reduction modulo \(p\): An elliptic curve \(E/\mathbb{Q}\) can be considered modulo each prime \(p\), yielding a curve \(\tilde{E}/\mathbb{F}_p\). Whether this reduction is smooth or has singularities determines the type of the curve at the place \(p\).
Local Galois representations: The restriction \(\rho|_{G_{\mathbb{Q}_p}}\) of a Galois representation to the local Galois group encodes the behaviour of the representation "at the place \(p\)". Mazur's deformation theory classifies representations by their local properties.
Semistability: An elliptic curve is called semistable if at every place \(p\) it has either good or multiplicative reduction. Wiles initially proved the Taniyama–Shimura conjecture only for semistable curves – which suffices for FLT because the Frey curve is semistable.

The \(p\)-adic numbers provide the "local lens" through which one examines algebraic objects prime by prime. Without them, Wiles' proof would be inconceivable.