8.1 p-adic numbers

P-adic numbers offer a fresh perspective on arithmetic properties, enabling local analysis of global problems in arithmetic geometry. They form a completion of rational numbers with respect to a p-adic metric, providing new insights into number-theoretic questions.

These numbers exhibit unique properties distinct from real numbers, crucial in arithmetic geometry. Understanding p-adic properties provides insights into the behavior of algebraic varieties over various fields, allowing for powerful local-global techniques in studying Diophantine equations.

Definition of p-adic numbers

• Arithmetic geometry utilizes p-adic numbers to study algebraic structures over number fields and finite fields
• p-adic numbers provide a different perspective on arithmetic properties, allowing for local analysis of global problems
• These numbers form a completion of the rational numbers with respect to a p-adic metric, offering new insights into number-theoretic questions

p-adic valuation

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• Measures the divisibility of a number by a prime p
• Defined as $v_p(x) = \max\{k \in \mathbb{Z} : p^k \text{ divides } x\}$ for non-zero rational numbers
• Extends to p-adic numbers through continuity
• Satisfies the strong triangle inequality $v_p(x + y) \geq \min(v_p(x), v_p(y))$

p-adic absolute value

• Derived from the p-adic valuation as $|x|_p = p^{-v_p(x)}$ for non-zero x
• Takes values in powers of p (1, p^{-1}, p^{-2}, ...)
• Satisfies the ultrametric inequality $|x + y|_p \leq \max(|x|_p, |y|_p)$
• Induces a metric on the rational numbers, defining the p-adic topology

Completion of rational numbers

• Constructs p-adic numbers as the completion of rational numbers with respect to the p-adic metric
• Analogous to the construction of real numbers as the completion of rationals under the usual metric
• Results in a complete metric space containing the rational numbers as a dense subset
• Yields a field of characteristic zero with a non-Archimedean absolute value

Properties of p-adic numbers

• p-adic numbers exhibit unique properties distinct from real numbers, crucial in arithmetic geometry
• These properties allow for powerful local-global techniques in studying Diophantine equations
• Understanding p-adic properties provides insights into the behavior of algebraic varieties over various fields

Non-Archimedean property

• Satisfies the strong triangle inequality $|x + y|_p \leq \max(|x|_p, |y|_p)$ for all x and y
• Results in unusual topological properties (every triangle is isosceles)
• Leads to the concept of p-adic discs, which are both open and closed
• Allows for simpler convergence criteria for series and power series

Ultrametric inequality

• Strengthens the triangle inequality to $|x + y|_p \leq \max(|x|_p, |y|_p)$ for all x and y
• Implies that p-adic balls are both open and closed (clopen)
• Ensures that any point inside a ball can serve as its center
• Simplifies many proofs and constructions in p-adic analysis

Hensel's lemma

• Provides a criterion for lifting solutions of polynomial congruences modulo p to solutions in p-adic integers
• States that if $f(a) \equiv 0 \pmod{p}$ and $f'(a) \not\equiv 0 \pmod{p}$, then there exists a unique p-adic integer x such that $f(x) = 0$ and $x \equiv a \pmod{p}$
• Generalizes to systems of polynomial equations
• Serves as a fundamental tool in p-adic analysis and algebraic number theory

p-adic expansions

• p-adic expansions provide a concrete representation of p-adic numbers, analogous to decimal expansions
• These expansions play a crucial role in computational aspects of arithmetic geometry
• Understanding p-adic expansions facilitates the study of p-adic functions and their properties

Canonical expansions

• Represents p-adic numbers as infinite series $\sum_{i=k}^{\infty} a_i p^i$ where $0 \leq a_i < p$
• Unique representation for each p-adic number
• Negative powers of p allowed for p-adic numbers not in Z_p
• Finite expansions correspond to rational numbers with denominators not divisible by p

Convergence of series

• Series $\sum a_n$ converges if and only if $\lim_{n \to \infty} |a_n|_p = 0$
• Convergence determined solely by the p-adic absolute values of the terms
• Allows for simpler convergence tests compared to real analysis
• Geometric series converge for |r|_p < 1, diverge for |r|_p ≥ 1

Arithmetic operations

• Addition performed digit-by-digit with carry, similar to base p arithmetic
• Multiplication uses distributive property and carries
• Division algorithm based on long division in base p
• Inversion of units utilizes the geometric series formula

p-adic integers

• p-adic integers form a fundamental object of study in arithmetic geometry
• They provide a bridge between local and global properties of algebraic structures
• Understanding p-adic integers is crucial for analyzing congruences and Diophantine equations

Ring structure

• Denoted as Z_p, forms a subring of Q_p
• Consists of p-adic numbers with non-negative valuation
• Complete discrete valuation ring with maximal ideal (p)
• Possesses unique factorization property

Units and non-units

• Units in Z_p are elements with p-adic absolute value 1
• Characterized by expansions with non-zero constant term
• Non-units form the unique maximal ideal (p)
• Unit group Z_p^* isomorphic to (Z/pZ)^* × (1 + pZ_p)

p-adic integers vs rational integers

• Z_p contains Z as a dense subring
• Every element of Z_p is a limit of a sequence of rational integers
• Z_p uncountable, while Z countable
• Z_p compact in p-adic topology, while Z discrete

p-adic fields

• p-adic fields extend the concept of p-adic numbers to algebraic extensions
• These fields play a crucial role in the study of local fields and their Galois theory
• Understanding p-adic fields is essential for analyzing the arithmetic of elliptic curves and higher-dimensional varieties

Field extensions

• Construct finite extensions of Q_p analogous to finite extensions of Q
• Unique unramified extension of Q_p for each degree n
• Totally ramified extensions correspond to Eisenstein polynomials
• Decomposition of extensions into unramified and totally ramified parts

Algebraic closure

• Algebraic closure of Q_p, denoted C_p, not complete
• Completion of algebraic closure yields the field of p-adic complex numbers
• C_p algebraically closed and complete, but not spherically complete
• Galois group of C_p over Q_p more complex than that of C over R

Ramification theory

• Studies how prime ideals decompose in field extensions
• Classifies primes as unramified, tamely ramified, or wildly ramified
• Ramification index and residue field degree product equals extension degree
• Crucial for understanding the behavior of primes in global fields

Applications in number theory

• p-adic methods provide powerful tools for solving classical number theory problems
• These applications demonstrate the interplay between local and global properties in arithmetic geometry
• Understanding p-adic applications enhances our ability to analyze Diophantine equations and arithmetic invariants

Local-global principle

• Relates the existence of solutions over Q to solutions over R and all Q_p
• Holds for quadratic forms (Hasse-Minkowski theorem)
• Fails in general (counterexamples include genus 1 curves)
• Motivates the study of Brauer groups and Shafarevich-Tate groups

Hasse-Minkowski theorem

• States that a quadratic form has a non-trivial rational solution if and only if it has a non-trivial solution in R and all Q_p
• Provides a complete criterion for the solvability of quadratic Diophantine equations
• Utilizes the concept of Hilbert symbol in its proof
• Generalizes to quadratic forms over number fields

p-adic zeta functions

• Analogues of Riemann zeta function defined over p-adic fields
• Encode information about arithmetic properties of varieties over finite fields
• Related to counting points on varieties modulo prime powers
• Play crucial role in Iwasawa theory and p-adic L-functions

p-adic analysis

• p-adic analysis extends classical real and complex analysis to the p-adic setting
• These techniques are essential for studying p-adic differential equations and p-adic cohomology theories
• Understanding p-adic analysis provides new insights into the behavior of algebraic varieties over p-adic fields

p-adic functions

• Continuous functions between p-adic spaces
• Analytic functions defined by convergent power series
• Weierstrass preparation theorem for p-adic analytic functions
• Newton polygons used to study zeros of p-adic analytic functions

p-adic integration

• Defines measures and integrals on p-adic spaces
• Haar measure on Q_p and its properties
• p-adic Fourier transform and its applications
• Connections to p-adic L-functions and special values

p-adic differential equations

• Studies solutions of differential equations over p-adic fields
• p-adic Frobenius structures and their role in p-adic cohomology
• Dwork's approach to counting points on varieties using p-adic differential equations
• Applications to p-adic periods and motives

Connections to arithmetic geometry

• p-adic methods provide essential tools for studying algebraic varieties over number fields and finite fields
• These connections highlight the interplay between local and global aspects of arithmetic geometry
• Understanding p-adic techniques enhances our ability to analyze arithmetic invariants of varieties

p-adic cohomology

• Cohomology theories for varieties over p-adic fields (crystalline, rigid, de Rham)
• Comparison theorems between various p-adic cohomology theories
• Applications to studying zeta functions and L-functions of varieties
• Connections to Galois representations and motives

p-adic periods

• Analogues of complex periods in the p-adic setting
• Relate de Rham and étale cohomology over p-adic fields
• Play crucial role in p-adic Hodge theory
• Applications to special values of L-functions and p-adic L-functions

p-adic uniformization

• Describes p-adic analytic spaces as quotients of p-adic symmetric spaces
• Analogous to complex uniformization of Riemann surfaces
• Applications to Shimura varieties and p-adic modular forms
• Connections to p-adic Langlands program

Computational aspects

• Computational techniques for p-adic numbers are crucial for applications in arithmetic geometry
• These methods allow for efficient calculation of p-adic invariants and solution of p-adic equations
• Understanding computational aspects enhances our ability to explore conjectures and generate examples in arithmetic geometry

p-adic algorithms

• Efficient algorithms for basic arithmetic operations in Q_p and Z_p
• Hensel lifting for solving polynomial equations over p-adic fields
• LLL algorithm and its p-adic variants for lattice reduction
• p-adic linear algebra algorithms (matrix inversion, determinants)

Precision and error analysis

• Concepts of absolute and relative p-adic precision
• Tracking precision loss in p-adic computations
• Interval arithmetic for p-adic numbers
• Strategies for maintaining and recovering precision in complex calculations

Software implementations

• Computer algebra systems with p-adic number support (PARI/GP, SageMath)
• Specialized libraries for p-adic computations (FLINT, p-adic)
• Implementations of p-adic algorithms in various programming languages
• Benchmarking and optimization techniques for p-adic computations