$p$-Adic Numbers

The real numbers arise by completing the rational numbers using the ordinary absolute value. The $p$-adic numbers arise by completing the rational numbers using the $p$-adic...

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A New Arithmetic Geometry

The real numbers arise by completing the rational numbers using the ordinary absolute value. The $p$-adic numbers arise by completing the rational numbers using the $p$-adic absolute value.

This produces a radically different geometry.

In the real numbers, powers of large integers grow without bound:

$$ 2^n\to\infty. $$

In the $2$-adic world,

$$ 2^n\to0. $$

The geometry is governed not by magnitude but by divisibility.

The resulting fields

$$ \mathbb{Q}_p $$

are central objects in modern number theory.

The $p$-Adic Absolute Value

Fix a prime $p$.

Every nonzero rational number can be written uniquely as

$$ x=p^k\frac{a}{b}, $$

where neither $a$ nor $b$ is divisible by $p$.

The $p$-adic valuation is

$$ v_p(x)=k. $$

The corresponding absolute value is

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

Thus divisibility by $p$ determines size.

Examples:

$$ |25|_5=5^{-2}=\frac1{25}, $$

$$ |7|_5=1. $$

Numbers divisible by high powers of $p$ become extremely small.

Metric and Distance

The $p$-adic absolute value defines a metric:

$$ d_p(x,y)=|x-y|_p. $$

This metric satisfies the ultrametric inequality:

$$ d_p(x,z) \le \max(d_p(x,y),d_p(y,z)). $$

Triangles therefore behave differently from Euclidean geometry.

Every triangle is isosceles, and often equilateral in the metric sense.

For example, if

$$ d_p(x,y)<d_p(y,z), $$

then

$$ d_p(x,z)=d_p(y,z). $$

Distances are dominated by the largest scale.

Completion of $\mathbb{Q}$

A sequence

$$ (x_n) $$

of rational numbers is $p$-adically Cauchy if

$$ |x_n-x_m|_p\to0 $$

as $n,m\to\infty$.

The completion of $\mathbb{Q}$ under this metric is the field

$$ \mathbb{Q}_p. $$

Elements of $\mathbb{Q}_p$ are limits of $p$-adic Cauchy sequences.

This construction parallels the construction of the real numbers from ordinary Cauchy sequences.

$p$-Adic Expansions

Every $p$-adic number has a series expansion

$$ a_0+a_1p+a_2p^2+\cdots, $$

where

$$ 0\le a_i<p. $$

Unlike decimal expansions, the series extends infinitely to the left in divisibility rather than in magnitude.

For example, in the $5$-adic numbers,

$$ \cdots +2\cdot5^3+4\cdot5^2+1\cdot5+3 $$

represents a valid $5$-adic number.

The coefficients encode divisibility information.

Negative Numbers in $p$-Adics

Negative integers acquire infinite expansions.

For example, in the $2$-adic numbers,

$$ -1 = 1+2+4+8+\cdots. $$

Indeed,

$$ 1+2+4+\cdots+2^n = 2^{n+1}-1, $$

and

$$ 2^{n+1}\to0 $$

in the $2$-adic metric.

Thus the limit equals

$$ -1. $$

This phenomenon illustrates how infinite sums behave differently in $p$-adic analysis.

The Ring of $p$-Adic Integers

The set

$$ \mathbb{Z}_p = {x\in\mathbb{Q}_p:|x|_p\le1} $$

is called the ring of $p$-adic integers.

Its elements have expansions

$$ a_0+a_1p+a_2p^2+\cdots $$

with no negative powers of $p$.

The unique maximal ideal is

$$ p\mathbb{Z}_p. $$

Every nonzero element can be written uniquely as

$$ p^k u, $$

where $u$ is a unit in $\mathbb{Z}_p$.

Thus $\mathbb{Z}_p$ behaves like a local version of the integers centered at the prime $p$.

Hensel Lemma

One of the most important tools in $p$-adic analysis is Hensel lemma.

Roughly speaking, it states that approximate polynomial solutions modulo powers of $p$ can often be lifted to genuine $p$-adic solutions.

For example, suppose

$$ f(a)\equiv0\pmod p $$

and

$$ f'(a)\not\equiv0\pmod p. $$

Then there exists a unique $p$-adic root near $a$.

Hensel lemma is a $p$-adic analogue of Newton iteration.

It allows local polynomial equations to be solved systematically.

Local Fields

The field

$$ \mathbb{Q}_p $$

is the basic example of a local field.

A local field is a complete field with respect to a discrete valuation and finite residue field.

Local fields support rich analytic and algebraic structures:

  • power series methods,
  • harmonic analysis,
  • Galois theory,
  • representation theory.

They are the local building blocks of global arithmetic.

Solving Equations Locally

Many Diophantine problems are first studied over

$$ \mathbb{Q}_p. $$

For example, one may ask whether

$$ x^2+y^2=z^2 $$

has nontrivial solutions over every $p$-adic field.

If an equation has no solution in some $\mathbb{Q}_p$, then it has no rational solution.

This principle is fundamental in arithmetic geometry.

Local-Global Principle

The interplay between rational numbers and all completions leads to the local-global philosophy.

A typical question asks:

If an equation has solutions in:

  • $\mathbb{R}$,
  • every $\mathbb{Q}_p$,

must it have a rational solution?

Sometimes the answer is yes, as in the Hasse-Minkowski theorem for quadratic forms.

Sometimes it is no.

Understanding these failures became one of the major themes of twentieth-century arithmetic geometry.

Haar Measure and Analysis

The field

$$ \mathbb{Q}_p $$

supports integration and harmonic analysis.

Compact subsets behave differently from real analysis:

  • closed balls are compact,
  • open balls are closed,
  • spaces become totally disconnected.

Nevertheless, Fourier analysis and measure theory still exist.

These structures are essential in automorphic forms and representation theory.

Structural Importance

$p$-adic numbers transformed number theory by introducing local analytic methods into arithmetic.

They now play central roles in:

  • local field theory,
  • Galois representations,
  • modular forms,
  • arithmetic geometry,
  • Iwasawa theory,
  • the Langlands program.

The $p$-adic world reveals arithmetic structure invisible from the viewpoint of ordinary real analysis.