Introduction to p-adic Analysis

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Mathematical Analysis

1 Algebraic Foundations

1.1 The p-adic Valuation

Definition 1.1 (p-adic valuation)

Let \(p\) be a prime number. For any rational number \(x \neq 0\), we can write \(x\) uniquely as \[x = p^k \cdot \frac{a}{b}\] where \(k \in \mathbb{Z}\), and \(a, b\) are integers with \(\gcd(a,p) = \gcd(b,p) = 1\). The p-adic valuation of \(x\) is defined as \(v_p(x) = k\). We set \(v_p(0) = +\infty\).

Properties of the p-adic valuation:

  1. \(v_p(xy) = v_p(x) + v_p(y)\)
  2. \(v_p(x + y) \geq \min\{v_p(x), v_p(y)\}\) with equality when \(v_p(x) \neq v_p(y)\)
  3. \(v_p(x) = +\infty\) if and only if \(x = 0\)

1.2 The p-adic Absolute Value

Definition 1.2 (p-adic absolute value)

The p-adic absolute value \(|\cdot|_p\) is defined by \[|x|_p = \begin{cases} 0 & \text{if } x = 0 \\ p^{-v_p(x)} & \text{if } x \neq 0 \end{cases}\]

Theorem 1.1 (Properties of p-adic absolute value)

The p-adic absolute value satisfies:

  1. \(|x|_p = 0\) if and only if \(x = 0\)
  2. \(|xy|_p = |x|_p |y|_p\)
  3. Strong triangle inequality (Ultrametric inequality): \(|x + y|_p \leq \max\{|x|_p, |y|_p\}\)

Properties 1 and 2 follow directly from the definition and properties of the valuation. For property 3:

If \(x = 0\) or \(y = 0\), the inequality is trivial. Otherwise, let \(v_p(x) = k\) and \(v_p(y) = \ell\). Then:

  • \(|x + y|_p = p^{-v_p(x+y)}\)
  • By property 2 of valuations, \(v_p(x + y) \geq \min\{k, \ell\}\)
  • Therefore, \(|x + y|_p = p^{-v_p(x+y)} \leq p^{-\min\{k,\ell\}} = \max\{p^{-k}, p^{-\ell}\} = \max\{|x|_p, |y|_p\}\) □
Corollary 1.1

In the p-adic metric, every triangle is isosceles. If \(|x|_p \neq |y|_p\), then \(|x + y|_p = \max\{|x|_p, |y|_p\}\).

1.3 Ostrowski’s Theorem

Theorem 1.2 (Ostrowski’s Theorem)

Every non-trivial absolute value on \(\mathbb{Q}\) is equivalent to either the usual absolute value \(|\cdot|_\infty\) or to some p-adic absolute value \(|\cdot|_p\) for a prime \(p\).

This theorem classifies all possible ways to measure “size” on the rational numbers, showing that the familiar absolute value and the p-adic absolute values are essentially the only possibilities.

1.4 Algebraic Consequences

Definition 1.3 (p-adic integers)

The ring of p-adic integers is defined as \[\mathbb{Z}_p = \{x \in \mathbb{Q} : |x|_p \leq 1\} = \{x \in \mathbb{Q} : v_p(x) \geq 0\}\]

Theorem 1.3

\(\mathbb{Z}_p\) is a local ring with unique maximal ideal \(\mathfrak{m} = p\mathbb{Z}_p = \{x \in \mathbb{Z}_p : |x|_p < 1\}\).

First, we show \(\mathbb{Z}_p\) is a ring. If \(x, y \in \mathbb{Z}_p\), then \(|x|_p \leq 1\) and \(|y|_p \leq 1\).

  • \(|xy|_p = |x|_p|y|_p \leq 1\), so \(xy \in \mathbb{Z}_p\)
  • \(|x + y|_p \leq \max\{|x|_p, |y|_p\} \leq 1\), so \(x + y \in \mathbb{Z}_p\)

The units of \(\mathbb{Z}_p\) are precisely \(\{x \in \mathbb{Z}_p : |x|_p = 1\}\), since if \(|x|_p = 1\), then \(x\) has inverse \(x^{-1}\) with \(|x^{-1}|_p = 1\). The non-units form the ideal \(\mathfrak{m} = \{x \in \mathbb{Z}_p : |x|_p < 1\}\), which is maximal since \(\mathbb{Z}_p/\mathfrak{m} \cong \mathbb{Z}/p\mathbb{Z}\) is a field. □

2 Analytic Foundations

2.1 Metric and Topological Properties

The p-adic absolute value induces a metric \(d_p(x,y) = |x - y|_p\) on \(\mathbb{Q}\). This metric has unusual properties compared to the usual Euclidean metric.

Theorem 2.1 (Ultrametric space properties)

The metric space \((\mathbb{Q}, d_p)\) satisfies:

  1. Every ball is both open and closed (clopen)
  2. If two balls intersect, one is contained in the other
  3. Every point in a ball is a center of that ball
  4. The space is totally disconnected

These follow from the strong triangle inequality. For example, for property 3: if \(z \in B_r(x) = \{y : |y - x|_p < r\}\), then for any \(w \in B_r(x)\), we have \(|w - z|_p \leq \max\{|w - x|_p, |x - z|_p\} < r\), so \(B_r(x) \subseteq B_r(z)\). Similarly, \(B_r(z) \subseteq B_r(x)\). □

2.2 Sequences and Convergence

Definition 2.1 (p-adic convergence)

A sequence \((x_n)\) in \(\mathbb{Q}\) converges p-adically to \(x\) if \(|x_n - x|_p \to 0\) as \(n \to \infty\).

Theorem 2.2

A sequence \((x_n)\) converges p-adically if and only if \(|x_{n+1} - x_n|_p \to 0\).

The “only if” direction is standard. For “if”: suppose \(|x_{n+1} - x_n|_p \to 0\). We show \((x_n)\) is Cauchy.

For any \(m > n\), we have: \[|x_m - x_n|_p = |x_m - x_{m-1} + x_{m-1} - x_{m-2} + \cdots + x_{n+1} - x_n|_p\]

By the ultrametric inequality applied repeatedly: \[|x_m - x_n|_p \leq \max\{|x_m - x_{m-1}|_p, |x_{m-1} - x_{m-2}|_p, \ldots, |x_{n+1} - x_n|_p\}\]

Since \(|x_{k+1} - x_k|_p \to 0\), given \(\epsilon > 0\), there exists \(N\) such that for all \(k \geq N\), \(|x_{k+1} - x_k|_p < \epsilon\). Therefore, for \(m > n \geq N\), \(|x_m - x_n|_p < \epsilon\), proving \((x_n)\) is Cauchy. □

2.3 Series and Power Series

Definition 2.2 (p-adic series)

A series \(\sum_{n=0}^{\infty} a_n\) converges p-adically if the sequence of partial sums converges p-adically.

Theorem 2.3 (p-adic convergence criterion)

A series \(\sum_{n=0}^{\infty} a_n\) converges p-adically if and only if \(|a_n|_p \to 0\) as \(n \to \infty\).

This is much stronger than the real case, where we only get convergence if terms go to zero, but the converse is not true.

Example: The geometric series \(\sum_{n=0}^{\infty} p^n\) diverges p-adically since \(|p^n|_p = p^{-n} \not\to 0\).

However, \(\sum_{n=0}^{\infty} p^n x^n\) converges for \(|x|_p < 1\) to \(\frac{1}{1-px}\).

2.4 p-adic Functions

Definition 2.3 (p-adic analytic function)

A function \(f: U \to \mathbb{Q}_p\) where \(U \subseteq \mathbb{Q}_p\) is open is called p-adic analytic if for every \(a \in U\), there exists a neighborhood \(V\) of \(a\) and a power series \(\sum_{n=0}^{\infty} c_n (x-a)^n\) that converges to \(f(x)\) for all \(x \in V\).

Key differences from real analysis:

  1. Radius of convergence: For \(\sum_{n=0}^{\infty} a_n x^n\), the radius is \(R = \frac{1}{\limsup_{n \to \infty} |a_n|_p^{1/n}}\)
  2. Behavior on boundary: p-adic power series often converge on their entire boundary
  3. Maximum principle: If \(f\) is analytic on a disk, then \(\max_{|x|_p \leq r} |f(x)|_p = \max_{|x|_p = r} |f(x)|_p\)

3 Completions

3.1 The Field of p-adic Numbers

The rational numbers \(\mathbb{Q}\) are not complete with respect to the p-adic metric. Just as we complete \(\mathbb{Q}\) with respect to the usual absolute value to get \(\mathbb{R}\), we can complete it with respect to \(|\cdot|_p\).

Definition 3.1 (p-adic numbers)

The field of p-adic numbers \(\mathbb{Q}_p\) is the completion of \(\mathbb{Q}\) with respect to the p-adic absolute value \(|\cdot|_p\).

Theorem 3.1 (Properties of \(\mathbb{Q}_p\))
  1. \(\mathbb{Q}_p\) is a complete metric space under the p-adic metric
  2. \(\mathbb{Q}_p\) is a field containing \(\mathbb{Q}\) as a dense subfield
  3. The p-adic absolute value extends uniquely to \(\mathbb{Q}_p\)
  4. \(\mathbb{Q}_p\) is locally compact

3.2 Construction via Cauchy Sequences

Elements of \(\mathbb{Q}_p\) can be represented as equivalence classes of Cauchy sequences in \(\mathbb{Q}\) under the p-adic metric.

Alternative representation: Every non-zero element \(x \in \mathbb{Q}_p\) can be written uniquely as: \[x = p^k \sum_{i=0}^{\infty} a_i p^i\] where \(k \in \mathbb{Z}\), \(a_i \in \{0, 1, 2, \ldots, p-1\}\), and \(a_0 \neq 0\).

3.3 The Ring of p-adic Integers

Definition 3.2 (Completion of p-adic integers)

The ring of p-adic integers \(\mathbb{Z}_p\) is the completion of \(\{x \in \mathbb{Q} : |x|_p \leq 1\}\), equivalently: \[\mathbb{Z}_p = \{x \in \mathbb{Q}_p : |x|_p \leq 1\}\]

Theorem 3.2 (Structure of \(\mathbb{Z}_p\))
  1. \(\mathbb{Z}_p\) is a compact, complete metric space
  2. \(\mathbb{Z}_p\) is a local ring with maximal ideal \(p\mathbb{Z}_p\)
  3. Every element of \(\mathbb{Z}_p\) can be written uniquely as \(\sum_{i=0}^{\infty} a_i p^i\) where \(a_i \in \{0, 1, \ldots, p-1\}\)
  4. \(\mathbb{Z}_p/p^n\mathbb{Z}_p \cong \mathbb{Z}/p^n\mathbb{Z}\)

3.4 Hensel’s Lemma

Theorem 3.3 (Hensel’s Lemma)

Let \(f(x) \in \mathbb{Z}_p[x]\) be a polynomial, and suppose \(a \in \mathbb{Z}_p\) satisfies:

  1. \(f(a) \equiv 0 \pmod{p}\)
  2. \(f'(a) \not\equiv 0 \pmod{p}\)

Then there exists a unique \(\alpha \in \mathbb{Z}_p\) such that \(f(\alpha) = 0\) and \(\alpha \equiv a \pmod{p}\).

This powerful lifting theorem allows us to solve polynomial equations in \(\mathbb{Q}_p\) by first solving them modulo \(p\) and then “lifting” the solutions.

Example: The equation \(x^2 = 2\) has solutions in \(\mathbb{Q}_7\) but not in \(\mathbb{Q}_3\), demonstrating that \(\mathbb{Q}_p\) depends crucially on the prime \(p\).

3.5 Applications and Further Directions

The theory of p-adic numbers connects to many areas of mathematics:

  • Number Theory: Studying Diophantine equations, local-global principles
  • Algebraic Geometry: p-adic varieties and rigid analytic spaces
  • Representation Theory: p-adic representations of Galois groups
  • Mathematical Physics: p-adic quantum mechanics and string theory

The completion process we’ve described here is fundamental to modern algebraic number theory and provides a powerful tool for understanding the arithmetic of rational numbers through “local” information at each prime.

3.6 References

For further reading on p-adic analysis, consult:

  • Koblitz, N. p-adic Numbers, p-adic Analysis, and Zeta-Functions
  • Robert, A. A Course in p-adic Analysis
  • Gouvêa, F. p-adic Numbers: An Introduction