9.2 Local-global principle
5 min read•august 21, 2024
The connects local properties of mathematical objects to their global behavior. It emerged from number theory studies in arithmetic geometry, providing powerful tools for solving Diophantine equations and analyzing on over number fields.
This principle investigates whether local solutions imply global solutions for mathematical problems. It utilizes to study local properties at each prime p and applies to various mathematical structures, including quadratic forms, Diophantine equations, and rational points on curves.
Origins and definition
- Local-global principle emerged from number theory studies in arithmetic geometry
- Connects local properties of mathematical objects to their global behavior
- Fundamental concept in understanding solutions to equations over various number fields
Historical development
Top images from around the web for Historical development
Novetats del fons: primera quinzena de juliol | Blog de la Biblioteca de Matemàtiques i Informàtica View original
Is this image relevant?
Dirac measure - Wikipedia View original
Is this image relevant?
On the local-global principle for isogenies of abelian surfaces | Selecta Mathematica View original
Is this image relevant?
Novetats del fons: primera quinzena de juliol | Blog de la Biblioteca de Matemàtiques i Informàtica View original
Is this image relevant?
Dirac measure - Wikipedia View original
Is this image relevant?
1 of 3
Top images from around the web for Historical development
Novetats del fons: primera quinzena de juliol | Blog de la Biblioteca de Matemàtiques i Informàtica View original
Is this image relevant?
Dirac measure - Wikipedia View original
Is this image relevant?
On the local-global principle for isogenies of abelian surfaces | Selecta Mathematica View original
Is this image relevant?
Novetats del fons: primera quinzena de juliol | Blog de la Biblioteca de Matemàtiques i Informàtica View original
Is this image relevant?
Dirac measure - Wikipedia View original
Is this image relevant?
1 of 3
- Introduced by Helmut Hasse in the 1920s for quadratic forms
- Extended by Ernst Kummer's work on cyclotomic fields in the 19th century
- Formalized by Emil Artin and John Tate in the 1950s through class field theory
Fundamental concepts
- Investigates whether local solutions imply global solutions for mathematical problems
- Utilizes p-adic numbers to study local properties at each prime p
- Applies to various mathematical structures (algebraic varieties, Galois representations)
Hasse principle vs local-global principle
- specifically refers to quadratic forms over
- Local-global principle encompasses broader mathematical contexts
- Both principles explore connections between local and global properties of mathematical objects
Applications in number theory
- Provides powerful tools for solving Diophantine equations in arithmetic geometry
- Enables analysis of rational points on algebraic varieties over number fields
- Connects local information at each prime to global properties of mathematical objects
Quadratic forms
- Applies to equations of the form over rational numbers
- Determines existence of non-trivial solutions using local-global principle
- Utilizes Hilbert symbol to analyze at each prime
Diophantine equations
- Studies polynomial equations with integer coefficients and integer solutions
- Applies local-global principle to determine existence of rational solutions
- Analyzes equations like () using this approach
Rational points on curves
- Investigates existence and distribution of rational points on algebraic curves
- Applies local-global principle to elliptic curves and higher genus curves
- Utilizes to study rational points on elliptic curves over number fields
Obstructions to local-global principle
- Identifies cases where local solutions do not guarantee global solutions
- Explores mathematical structures that prevent local-global principle from holding
- Provides insights into limitations of arithmetic geometry techniques
Brauer group
- Measures obstruction to local-global principle for rational points on varieties
- Defined as group of equivalence classes of central simple algebras over a field
- Utilizes cohomological methods to study
Adelic points
- Considers points on varieties over adele ring of a number field
- Analyzes local-global principle using adelic topology
- Provides framework for studying rational points on varieties over global fields
Shafarevich-Tate group
- Measures failure of local-global principle for abelian varieties
- Defined as group of homogeneous spaces with no rational points
- Conjectured to be finite for elliptic curves over number fields
Examples and counterexamples
- Illustrates applications and limitations of local-global principle in arithmetic geometry
- Provides concrete cases to understand theoretical concepts
- Demonstrates importance of studying both positive and negative results
Positive results
- Hasse-Minkowski theorem for quadratic forms over rational numbers
- Weak approximation for linear algebraic groups
- Local-global principle for norm equations in cyclic extensions
Negative results
- Selmer's example of a cubic surface violating local-global principle
- Reichardt's counterexample for genus 1 curves
- Failure of local-global principle for certain K3 surfaces
Famous counterexamples
- Selmer's cubic surface
- Lind-Reichardt curve
- Cassels-Guy counterexample for simultaneous representation by quadratic forms
Methods of proof
- Develops techniques to analyze local-global principle in arithmetic geometry
- Combines algebraic, analytic, and geometric approaches
- Utilizes advanced mathematical tools from various branches of mathematics
Local methods
- Employs p-adic analysis to study local properties at each prime
- Utilizes Hensel's lemma for lifting solutions from finite fields to p-adic fields
- Applies local-global principle to quadratic forms using Hilbert symbol
Global methods
- Utilizes algebraic number theory techniques for global analysis
- Applies class field theory to study Galois extensions of number fields
- Employs analytic methods (circle method, sieve methods) for certain Diophantine equations
Cohomological approaches
- Utilizes étale cohomology to study arithmetic properties of varieties
- Applies to analyze obstructions to local-global principle
- Employs derived category methods for studying derived Brauer-Manin obstruction
Generalizations and extensions
- Expands local-global principle beyond classical number theory contexts
- Explores applications in broader mathematical settings
- Develops new tools and techniques for arithmetic geometry
Function fields
- Applies local-global principle to function fields of curves over finite fields
- Studies arithmetic of elliptic curves over function fields
- Investigates analogues of classical results (Mordell-Weil theorem) in function field setting
Higher-dimensional varieties
- Extends local-global principle to algebraic varieties of dimension greater than 1
- Analyzes rational points on K3 surfaces and Calabi-Yau varieties
- Investigates local-global principle for homogeneous spaces of linear algebraic groups
Adelic spaces
- Generalizes local-global principle to adelic points on varieties
- Studies topology of adelic spaces associated to algebraic varieties
- Investigates strong approximation property for algebraic groups over adeles
Connections to other areas
- Demonstrates interdisciplinary nature of local-global principle in mathematics
- Illustrates connections between arithmetic geometry and other mathematical fields
- Provides framework for applying diverse mathematical techniques to number theory problems
Class field theory
- Utilizes local-global principle in studying abelian extensions of number fields
- Applies reciprocity laws to analyze local and global properties of field extensions
- Connects local class field theory to global class field theory through idelic formulation
Galois cohomology
- Employs cohomological methods to study local-global principle
- Utilizes Galois cohomology groups to analyze obstructions to local-global principle
- Applies Tate-Shafarevich groups to study failure of local-global principle for abelian varieties
Algebraic groups
- Investigates local-global principle for linear algebraic groups
- Studies rational points on homogeneous spaces of algebraic groups
- Applies local-global principle to problems in arithmetic of algebraic groups
Recent developments
- Explores cutting-edge research in local-global principle and arithmetic geometry
- Identifies current challenges and unsolved problems in the field
- Provides insights into potential future directions for research
Advances in the 21st century
- Development of derived Brauer-Manin obstruction by Alexei Skorobogatov
- Progress on Brumer-Kramer conjecture for abelian surfaces
- Advances in understanding local-global principle for K3 surfaces
Open problems
- Birch and Swinnerton-Dyer conjecture for elliptic curves
- Finiteness of Shafarevich-Tate group for abelian varieties
- Local-global principle for rational points on higher genus curves
Future directions
- Investigating local-global principle in non-commutative arithmetic geometry
- Applying homotopy theory techniques to study local-global principle
- Exploring connections between local-global principle and arithmetic dynamics
Key Terms to Review (19)
Algebraic Varieties: Algebraic varieties are fundamental objects in algebraic geometry, defined as the solution sets of systems of polynomial equations. They serve as geometric manifestations of algebraic equations and can be classified into several types, including affine varieties and projective varieties, each possessing unique properties. These varieties are essential for studying the interplay between geometry and algebra, connecting to concepts like local-global principles, obstructions, and comparison theorems.
Brauer-Manin obstruction: The Brauer-Manin obstruction is a method used to understand the solvability of equations over global fields, particularly in the context of algebraic varieties. This concept connects local and global properties of varieties, illustrating how certain local conditions can prevent a global solution, even when solutions exist locally. It highlights the interplay between the Brauer group of a variety and rational points, contributing to broader themes such as the local-global principle, Hasse principle, and weak approximation.
Chabauty's method: Chabauty's method is a powerful technique in number theory used to determine the rational points on certain curves, particularly when the genus of the curve is greater than one. This method leverages the connection between algebraic geometry and arithmetic, utilizing p-adic analysis to count the number of rational solutions effectively. By analyzing the curve's properties over p-adic fields, Chabauty’s method can often provide insight into which points are rational, helping to bridge local information with global rationality.
Cohomological obstructions: Cohomological obstructions refer to the elements in cohomology groups that prevent certain geometric or algebraic properties from holding true for a space or a scheme. These obstructions can arise in various contexts, such as when trying to lift solutions locally to a global solution, thus highlighting the discrepancies between local and global properties.
David Hilbert: David Hilbert was a prominent German mathematician who made significant contributions to various areas of mathematics, including algebraic number theory and algebraic geometry. His work laid the groundwork for many modern theories and techniques, influencing fields such as arithmetic geometry and number theory, which explore the relationships between algebraic structures and their geometric interpretations.
Descent Theory: Descent theory is a method in algebraic geometry that studies the properties of schemes by relating them to simpler schemes through a process known as descent. It connects local properties of varieties over various base fields to global properties, allowing for a better understanding of rational points and morphisms, which are crucial in different contexts such as the study of abelian varieties, surfaces, and higher-dimensional varieties.
Fermat's Last Theorem: Fermat's Last Theorem states that there are no three positive integers $a$, $b$, and $c$ that satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than 2. This theorem is famously known for remaining unproven for over 350 years until it was finally resolved by Andrew Wiles in 1994, establishing a deep connection with modular forms and elliptic curves, which ties into several advanced concepts in number theory.
Galois Cohomology: Galois cohomology is a mathematical framework that studies the relationships between Galois groups and the field extensions they act upon, using cohomological methods. This approach connects algebraic structures, like group cohomology, with number theory and arithmetic geometry, helping to understand phenomena such as the local-global principle and obstructions to finding rational points on varieties.
Global Condition: A global condition refers to a state of a mathematical object or property that is true across all relevant fields or contexts, not just within local subsets. This concept is crucial in arithmetic geometry, particularly when considering whether certain properties or solutions hold universally, linking local observations to a broader, global perspective.
Hasse Principle: The Hasse Principle is a concept in number theory and arithmetic geometry that asserts that a global solution to a Diophantine equation exists if and only if solutions exist locally in all completions of the field, including the p-adic numbers and the real numbers. This principle connects various mathematical structures and offers insight into when we can find rational points on algebraic varieties.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his significant contributions to algebraic geometry, topology, and number theory. His work has deeply influenced various fields within mathematics, particularly in relation to the development of modern concepts and conjectures surrounding arithmetic geometry.
Local condition: A local condition refers to a property or characteristic that holds true in a localized setting, often in the context of a particular point or neighborhood in a geometric space. Understanding local conditions is crucial for determining global properties through local-global principles, which connect local behavior at various points to a broader, more global perspective.
Local solvability: Local solvability refers to the existence of solutions to mathematical equations or problems within a localized context, often over specific fields or local rings. This concept is crucial in understanding whether a global solution exists by examining solutions in smaller, localized settings. Local solvability plays a significant role in connecting local conditions to global properties, which is essential when considering the local-global principle and the Hasse principle.
Local-Global Principle: The local-global principle is a concept in number theory and algebraic geometry that asserts a property holds globally if it holds locally at all places. This idea connects local solutions, examined through local fields, to the existence of global solutions on varieties or more general schemes, showcasing the interplay between local and global perspectives.
Mordell-Weil Theorem: The Mordell-Weil Theorem states that the group of rational points on an elliptic curve over a number field is finitely generated. This fundamental result connects the theory of elliptic curves with algebraic number theory, revealing the structure of rational solutions and their relationship to torsion points and complex multiplication.
Obstruction Theory: Obstruction theory is a framework in algebraic geometry and topology that deals with the conditions under which certain geometric or algebraic structures can be lifted from a lower-dimensional space to a higher-dimensional space. This concept is particularly important in understanding the behavior of solutions to equations and their existence in different settings, like local versus global properties.
P-adic numbers: p-adic numbers are a system of numbers used in number theory, which extend the concept of the usual integers and rational numbers. They provide a way to study congruences and properties of numbers at a prime level, allowing mathematicians to analyze problems from local and global perspectives. The p-adic numbers are essential in various areas such as geometry, forms, and the interplay between algebraic and analytic structures.
Rational numbers: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. They include integers, finite decimals, and repeating decimals, making them an essential part of arithmetic and number theory. Understanding rational numbers is critical for exploring concepts such as local-global principles, where solutions to equations are examined in various number systems.
Rational Points: Rational points are solutions to equations that can be expressed as fractions of integers, typically where the coordinates are in the form of rational numbers. These points are crucial in the study of algebraic varieties, especially in understanding the solutions over different fields, including the rational numbers, which can reveal deeper properties of the geometric objects involved.