11.3 Local-global principle
4 min read•july 30, 2024
The is a powerful tool in algebraic number theory. It lets us study global properties of mathematical objects by looking at their local behavior in different number fields. This approach simplifies complex problems and provides insights into equations, , and algebraic structures.
While the principle often works, it's not foolproof. Famous counterexamples exist, pushing mathematicians to develop more refined methods. Understanding when and why the principle fails deepens our grasp of number theory and opens new research avenues in this fascinating field.
The local-global principle
Fundamental concepts and significance
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- Local-global principle asserts properties of mathematical objects defined over a can be determined by examining properties over all of the field
- Global field typically refers to a number field in algebraic number theory
- Completions include for all prime numbers p and the real numbers
- Property holding locally everywhere (for all completions) implies it holds globally (over the original field)
- Reduces global problems to local problems, often easier to solve
- Provides powerful tool for understanding behavior of algebraic structures across different number fields
- Applies to many important cases, though known counterexamples exist where does not imply
Applications in algebraic number theory
- Determines existence of solutions to
- Studies properties of quadratic forms
- Analyzes on ()
- Investigates local-global principles for more general algebraic structures ()
- Examines and its relationship to local-global phenomena
- Explores connections with and
Solving Diophantine equations
Applying the local-global principle
- Check for solutions in all completions of rational numbers to determine existence of rational solutions
- Examine solutions modulo prime powers (p-adic fields) and in real numbers
- Solutions existing in all local fields suggest existence of global rational solution (not guaranteed for all equations)
- Use for p-adic fields and continuity arguments for real solutions
- Break down problem into more manageable local cases, simplifying solving process
- Apply to quadratic equations in two or more variables (classic example where principle often succeeds)
- Utilize additional techniques (descent methods, study of Brauer groups) when local-global principle fails
Specific methods and examples
- Solve congruences modulo increasing powers of primes to find p-adic solutions
- Use to combine local solutions
- Apply for quadratic forms
- Analyze elliptic curves using local-global methods ()
- Study higher degree equations (, equations of degree ≥ 4)
- Implement computational methods to check local solvability efficiently
- Explore connections with modular forms and Galois representations ()
The Hasse principle
Quadratic forms and the Hasse-Minkowski theorem
- states quadratic form over rationals has non-trivial solution if and only if it has non-trivial solution over real numbers and p-adic numbers for all primes p
- Hasse-Minkowski theorem confirms validity of Hasse principle for quadratic forms in any number of variables over number fields
- Apply Hilbert symbol to study local behavior of quadratic forms
- Use product formula for Hilbert symbols to connect local and global information
- Analyze genus theory of quadratic forms using local-global principles
- Extend results to quadratic forms over function fields
- Study spinor genus and its relationship to the Hasse principle
Applications to elliptic curves
- Study existence of rational points on elliptic curves using Hasse principle
- Relate Hasse principle to
- Determine over number fields by analyzing local information
- Apply Hasse principle in elliptic curve cryptography (point counting, discrete logarithm problem)
- Investigate Mordell-Weil theorem and its connections to the Hasse principle
- Explore and their role in understanding the failure of the Hasse principle
- Study and their relationship to local-global principles for elliptic curves
Limitations of the local-global principle
Famous counterexamples
- Ernst Selmer's 1951 cubic curve counterexample:
- Birch and Swinnerton-Dyer's 1975 counterexample:
- Cassels and Guy's 1966 diagonal cubic surface:
- Poonen's 2009 counterexample of every odd degree ≥ 3
- Skorobogatov's 1999 bielliptic surface counterexample
- Brumer's 1978 counterexample involving a genus 1 curve
Advanced concepts and developments
- Non-trivial often relates to failure of Hasse principle
- refines original principle to account for some failures
- connects to failures of Hasse principle for elliptic curves
- Develop new tools in arithmetic geometry to study rational points on varieties
- Investigate (Manin's conditional algorithm) for analyzing failures
- Study obstruction sets (e.g., Brauer-Manin set, étale-Brauer set) to refine local-global principles
- Explore connections with Diophantine approximation and unlikely intersections
Key Terms to Review (32)
Adelic methods: Adelic methods are techniques in number theory that involve the use of adèles, which are a type of mathematical object that allows for the study of numbers and their properties across various local and global contexts. These methods facilitate understanding the relationships between local solutions to equations and their corresponding global solutions, often providing a framework to tackle problems related to the local-global principle.
Algebraic Varieties: Algebraic varieties are geometric objects that are the solution sets of systems of polynomial equations. They play a crucial role in algebraic geometry, linking algebraic equations with geometric shapes. Understanding these varieties helps connect various areas of mathematics, particularly through their local and global properties, which are essential for studying concepts like the local-global principle.
Birch and Swinnerton-Dyer Conjecture: The Birch and Swinnerton-Dyer Conjecture is a fundamental hypothesis in number theory that relates the number of rational points on an elliptic curve to the behavior of its associated L-function at a specific point. It suggests that the rank of the group of rational points on the elliptic curve can be determined by analyzing the vanishing of the L-function at a critical value, particularly when it equals zero. This conjecture connects deep aspects of algebraic geometry, number theory, and complex analysis.
Brauer-Manin Obstruction: The Brauer-Manin obstruction is a concept in algebraic geometry and number theory that explains how certain varieties may fail to have rational points due to the existence of nontrivial elements in the Brauer group. This idea connects local and global aspects of solutions, where a variety might appear to have points locally but not globally. The obstruction helps in understanding the discrepancies between local solvability and global solvability, particularly within the framework of the local-global principle.
Chinese Remainder Theorem: The Chinese Remainder Theorem is a fundamental result in number theory that provides a way to solve systems of simultaneous congruences with pairwise coprime moduli. This theorem states that if you have multiple congruences, you can find a unique solution modulo the product of these moduli. It connects to prime ideals and maximal ideals by showing how the structure of integers can be analyzed through these ideals, and it relates to the local-global principle by illustrating how local solutions can lead to a global solution.
Class Field Theory: Class field theory is a branch of algebraic number theory that studies the relationships between abelian extensions of number fields and the ideal class groups of those fields. It provides a powerful framework for understanding how number fields can be related to one another through their Galois groups, while also linking local properties at primes with global properties of the number field.
Completions: Completions refer to a method of constructing a new mathematical object from an existing one by adding 'limits' or 'points' to make it more complete in terms of certain properties, like convergence. This concept is particularly important when examining local properties of mathematical structures, allowing us to connect these local perspectives with global characteristics, and is essential in the study of algebraic number theory.
Cubic Surfaces: Cubic surfaces are algebraic surfaces defined by polynomial equations of degree three in projective space. These surfaces have a rich structure and can be studied through their geometric properties and intersections, making them important in various areas of mathematics, including algebraic geometry and number theory.
Diophantine equations: Diophantine equations are polynomial equations that seek integer solutions. Named after the ancient Greek mathematician Diophantus, these equations often reflect deep number-theoretic properties and have been central in various mathematical investigations, including the study of unique factorizations and local-global principles. They serve as a bridge connecting number theory to algebraic structures and have significant implications in understanding reciprocity laws and the classification of solutions to famous problems.
Elliptic Curves: Elliptic curves are smooth, projective curves defined over a field that have a specific algebraic structure, typically expressed in the form $$y^2 = x^3 + ax + b$$ where the curve has no singular points. These curves play a significant role in number theory, particularly in the study of Diophantine equations and have profound implications in various areas such as cryptography and class field theory.
étale-Brauer Hasse principle: The étale-Brauer Hasse principle is a statement in algebraic number theory that connects the existence of certain algebraic objects, known as Azumaya algebras, to their local conditions at various places of a field. This principle suggests that if an Azumaya algebra is locally trivial at all completions of a field, then it is globally trivial. It is crucial in understanding how local-global principles apply to the study of Brauer groups and cohomology.
Galois Cohomology: Galois cohomology is a branch of mathematics that studies the relationship between algebraic structures and their symmetries, focusing on the actions of Galois groups on cohomological objects. It provides tools to investigate local-global principles, allowing mathematicians to relate local properties of algebraic varieties or number fields to global properties through the lens of Galois theory. This connection plays a crucial role in number theory and algebraic geometry.
Global Field: A global field is a field that is either a number field or a function field over a finite field. These fields are significant in number theory and algebra because they allow mathematicians to study the properties of numbers and their relationships in a more unified way. Global fields serve as the foundation for various important concepts, linking together aspects such as class numbers, strong approximation, local-global principles, idele groups, and adele rings.
Global solvability: Global solvability refers to the condition where a mathematical equation or system of equations has solutions that can be found not just locally (in the neighborhood of a point) but also across the entire space considered, such as over all rational numbers or integers. This concept is deeply connected to the local-global principle, which states that if a problem can be solved locally at every completion of a number field, then it should also have a solution globally. Understanding global solvability is crucial in determining whether equations can be solved in broader contexts based on their local properties.
Group Cohomology: Group cohomology is a mathematical tool that studies the extensions and representations of groups by analyzing how group actions affect cochains, leading to insights about the structure and properties of the group itself. It generalizes the notion of cohomology from algebraic topology to group theory, allowing for a deeper understanding of how groups can be represented and related to topological spaces. This concept plays a crucial role in connecting local properties of groups to global phenomena, highlighting the local-global principle.
Hasse Principle: The Hasse Principle is a concept in number theory that states a polynomial equation can have solutions in rational numbers if and only if it has solutions in the p-adic numbers for all primes p. This principle helps connect local properties of equations with global properties, making it a crucial idea in understanding when equations can be solved over the rationals.
Hasse Principle in Cryptography: The Hasse Principle in cryptography refers to a fundamental concept that connects local and global properties of algebraic objects, specifically in relation to the solvability of equations over fields. This principle emphasizes that if an equation has solutions in all local completions of a field, then it should also have a solution in the global field. This connection is crucial in the study of elliptic curves and other algebraic structures used in cryptographic systems.
Hasse-Minkowski Theorem: The Hasse-Minkowski Theorem states that a quadratic form over the rational numbers is representable by integers if and only if it is representable over the p-adic numbers for all primes p, and also over the real numbers. This theorem links local conditions, where one checks the form's representability over local fields, to a global condition concerning its representability over rational numbers, reflecting the local-global principle in number theory.
Height Functions: Height functions are tools used in number theory to measure the complexity or size of algebraic numbers or points in projective space. These functions help in understanding the distribution of rational points and are crucial for analyzing the local-global principles, which connect local properties of solutions to global ones.
Hensel's Lemma: Hensel's Lemma is a fundamental result in p-adic analysis that provides a criterion for lifting solutions of polynomial equations from the residue field to the p-adic integers. It connects the concept of p-adic numbers with algebraic equations, allowing us to find roots in a more refined p-adic setting. This lemma is crucial for understanding local properties of algebraic equations and plays a key role in various advanced concepts such as completions, strong approximations, and local-global principles.
Local solvability: Local solvability refers to the property of an equation or a mathematical problem being solvable in the context of local fields or at specific primes. It plays a crucial role in understanding whether a global solution exists by examining the problem's behavior at these localized settings. The idea is that if a problem can be solved locally, it may lead to insights about its solvability in a broader sense, highlighting the connection between local and global properties of equations.
Local-global principle: The local-global principle is a concept in number theory that asserts a statement about numbers holds true globally if it holds true locally in various completions of the number field. Essentially, if a solution exists in all local settings (such as the p-adic numbers and real numbers), then there exists a corresponding global solution in the rational numbers. This principle connects local properties to global existence, making it a foundational idea in understanding how different number systems relate to each other.
Mordell-Weil Theorem: The Mordell-Weil Theorem states that for an elliptic curve defined over a number field, the group of rational points on the curve is finitely generated. This means that the set of rational solutions can be expressed as a finite direct sum of a torsion subgroup and a free abelian group of finite rank, providing crucial insight into the structure of the solutions to elliptic curves.
Obstruction Theory: Obstruction theory is a framework in algebraic number theory that investigates the conditions under which local properties of mathematical objects can be used to infer global properties. It addresses the challenges faced when trying to extend local solutions, typically found in a local field, to a global solution over a number field. This theory is crucial for understanding how local and global properties interact, especially when considering rational points on algebraic varieties.
P-adic fields: p-adic fields are number fields that extend the concept of p-adic numbers, which allow us to work with numbers in a way that emphasizes their divisibility by a prime number p. These fields provide a different perspective on number theory and are instrumental in understanding local properties of numbers, leading to connections with global arithmetic properties, such as those involved in the local-global principle.
Quadratic Forms: Quadratic forms are homogeneous polynomial expressions of degree two in multiple variables, typically represented in the form $$Q(x_1, x_2, ext{...}, x_n) = a_1x_1^2 + a_2x_2^2 + ... + a_nx_n^2 + ext{terms with cross products}$$. They play a crucial role in number theory, particularly in understanding the representation of integers as sums of squares and in connecting local properties with global solutions.
Rank of Elliptic Curves: The rank of an elliptic curve is an important integer that measures the number of independent rational points on the curve. It helps in understanding the structure of the group of rational points and plays a crucial role in connecting local properties of the curve with global ones, particularly through concepts like the local-global principle. The rank provides insight into whether certain equations have solutions in rational numbers and can indicate how many solutions exist.
Rational Points: Rational points are specific solutions to equations that take the form of ordered pairs of rational numbers, typically represented as $(x,y) \in \mathbb{Q}^2$. They play a crucial role in various areas of number theory and geometry, particularly when discussing the solvability of equations over the rational numbers. The importance of rational points is highlighted in many results that connect local conditions (solutions in the p-adic sense) with global conditions (solutions over the rationals), revealing deep insights about the structure of mathematical objects.
Selmer Groups: Selmer groups are mathematical structures that arise in the study of abelian varieties and their rational points. They serve as a bridge between local and global properties of these varieties, playing a key role in the local-global principle, which asserts that the existence of solutions to equations over global fields can often be inferred from their existence over local fields. Selmer groups help to analyze the obstructions to finding rational points, making them crucial in understanding the arithmetic of elliptic curves and other algebraic varieties.
Serre's Modularity Conjecture: Serre's Modularity Conjecture is a significant hypothesis in number theory which proposes that every rational elliptic curve is modular, meaning it can be associated with a modular form. This conjecture connects the realms of algebraic geometry and number theory, suggesting a deep link between the properties of elliptic curves and modular forms, which are essential in understanding solutions to Diophantine equations and properties of L-functions.
Tate-Shafarevich Group: The Tate-Shafarevich group is an important algebraic structure associated with an elliptic curve and an algebraic number field, capturing information about the failure of the local-global principle for the curve. It consists of isomorphism classes of torsors under the elliptic curve that do not have a rational point but have points over every completion of the number field. This group serves as a measure of how much the solutions to equations can differ between local and global perspectives.
Transcendental methods: Transcendental methods are techniques used in number theory that help to prove the irrationality or transcendence of certain numbers. These methods often involve sophisticated arguments, such as using properties of algebraic functions and forms, and can lead to significant results regarding the relationships between algebraic and transcendental numbers.