9.5 Weak approximation
6 min read•august 21, 2024
is a fundamental concept in arithmetic geometry, connecting local and global properties of algebraic varieties. It describes how closely on varieties can approximate solutions in all completions of a number field simultaneously.
The weak approximation theorem provides conditions for rational points to be dense in of a variety. This has significant implications for understanding the distribution of rational solutions to and studying arithmetic properties of algebraic varieties.
Definition of weak approximation
- Fundamental concept in arithmetic geometry connecting local and global properties of algebraic varieties
- Describes ability to approximate rational points on varieties using local data from various completions
- Crucial for understanding distribution of rational solutions to Diophantine equations
Adelic points vs rational points
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- Adelic points represent solutions in all completions of a number field simultaneously
- Rational points embed diagonally into the set of adelic points
- Weak approximation measures how closely rational points approximate adelic points
- Topological closure of rational points in adelic space key to weak approximation property
Relation to local-global principle
- Weak approximation strengthens for rational points on varieties
- Local-global principle states existence of local solutions implies global solution
- Weak approximation additionally requires approximation of local solutions by global ones
- Failure of weak approximation can occur even when local-global principle holds
Weak approximation theorem
- Central result in arithmetic geometry connecting local and global properties of algebraic varieties
- Provides conditions under which rational points are dense in the adelic points of a variety
- Implications for existence and distribution of rational solutions to Diophantine equations
Statement and significance
- For a smooth, geometrically connected variety X over a number field K, weak approximation holds if rational points are dense in adelic points
- Density measured in product topology of all non-archimedean and archimedean completions of K
- Significance lies in ability to approximate solutions in all completions simultaneously by rational points
- Provides powerful tool for studying rational points on varieties over number fields
Proof outline
- Utilizes theorem for
- Reduces problem to studying weak approximation for homogeneous spaces
- Employs fibration method to reduce to simpler varieties
- Involves analysis of Galois cohomology groups and their behavior under field extensions
- Concludes with application of Hilbert's Nullstellensatz in suitable context
Applications in arithmetic geometry
- Weak approximation plays crucial role in studying rational points on algebraic varieties
- Provides insights into distribution and density of solutions to Diophantine equations
- Connects local and global properties of varieties over number fields
Rational points on varieties
- Weak approximation used to prove existence of rational points on certain varieties
- Helps determine density of rational points in real or p-adic topologies
- Applied to study rational points on quadrics, cubic surfaces, and other low-degree varieties
- Provides criteria for when local solubility implies global solubility for systems of equations
Diophantine equations
- Weak approximation aids in solving Diophantine equations over number fields
- Used to construct rational solutions approximating given local solutions
- Applies to equations defining varieties with weak approximation property
- Helps determine when system of Diophantine equations has infinitely many solutions
Obstructions to weak approximation
- Identify reasons why weak approximation may fail for certain varieties
- Understanding obstructions crucial for developing refined versions of weak approximation
Brauer-Manin obstruction
- Arises from Brauer group of an algebraic variety
- Measures failure of weak approximation due to global constraints on local solutions
- Computed using cohomological methods and étale cohomology
- Can explain some cases where local-global principle fails for rational points
Descent obstruction
- Related to Galois cohomology and descent theory for algebraic varieties
- Arises from non-trivial torsors under linear algebraic groups
- Can detect failures of weak approximation not explained by
- Involves study of Selmer groups and their relation to rational points
Weak approximation for algebraic groups
- Study of weak approximation property for specific classes of algebraic varieties
- Focuses on varieties with additional group structure
Linear algebraic groups
- Weak approximation holds for connected linear algebraic groups over number fields
- Proof uses strong approximation theorem for simply connected semisimple groups
- Applications to studying rational points on homogeneous spaces
- Connections to class field theory and adelic quotients of algebraic groups
Abelian varieties
- Weak approximation fails in general for
- Obstruction related to finiteness of Tate-Shafarevich group
- Study involves deep results from arithmetic of elliptic curves and higher-dimensional abelian varieties
- Connections to BSD conjecture and Mordell-Weil theorem
Strong approximation
- Stronger version of weak approximation with additional constraints
- Crucial for understanding arithmetic properties of algebraic groups
Comparison with weak approximation
- Strong approximation requires density in adelic points with restricted set of places
- Holds for simply connected semisimple algebraic groups over number fields
- Implies weak approximation but converse does not hold in general
- Applications to studying integral points on varieties
Examples and counterexamples
- Strong approximation holds for SL_n over number fields
- Fails for GL_n due to determinant condition
- Tori provide examples where strong approximation fails but weak approximation holds
- Connections to class number problems in algebraic number theory
Weak approximation in function fields
- Study of weak approximation for varieties defined over
- Analogous to number field case but with important differences
Analogies with number fields
- Function field analogue of adeles and ideles
- Similar formulation of weak approximation property for varieties
- Connections to zeta functions and L-functions of varieties over function fields
- Analogues of and its failures in function field setting
Specific characteristics
- Presence of generic point in function field case
- Behavior of weak approximation under base change to finite extensions
- Role of Frobenius endomorphism in studying rational points
- Connections to Grothendieck's section conjecture in anabelian geometry
Computational aspects
- Development of algorithms and computational methods for studying weak approximation
- Important for practical applications and testing conjectures
Algorithms for testing weak approximation
- Methods for computing Brauer-Manin obstruction
- Algorithms for checking local solubility at finite set of places
- Techniques for constructing adelic points approximating given local solutions
- Implementation of descent methods for detecting obstructions
Complexity considerations
- Analysis of time and space complexity for weak approximation algorithms
- Dependence on height of coefficients and degree of defining equations
- Comparison of different algorithmic approaches (algebraic vs analytic)
- Limitations and challenges in high-dimensional cases
Recent developments
- Ongoing research and new results in the study of weak approximation
- Connections to other areas of mathematics and number theory
Open problems
- Generalizations of weak approximation to more general arithmetic schemes
- Refined versions of weak approximation taking into account known obstructions
- Effective results on density of rational points satisfying weak approximation
- Connections to dynamical systems and ergodic theory on adelic spaces
Connections to other areas
- Interactions with Langlands program and automorphic forms
- Applications to arithmetic statistics and counting problems
- Connections to non-abelian reciprocity laws and anabelian geometry
- Relevance to arithmetic dynamics and potential theory on adelic spaces
Historical context
- Development of weak approximation concept in number theory and arithmetic geometry
- Evolution of understanding and connections to other areas
Origins of the concept
- Roots in early 20th century work on Diophantine approximation
- Contributions of Hasse, Weil, and Kneser in formulating weak approximation
- Connections to development of adelic methods in number theory
- Influence of algebraic groups theory on study of weak approximation
Evolution of understanding
- Refinements and generalizations of weak approximation concept over time
- Discovery and classification of obstructions to weak approximation
- Impact of étale cohomology and Grothendieck's work on modern formulations
- Recent connections to arithmetic dynamics and ergodic theory on adelic spaces
Key Terms to Review (27)
Abelian varieties: Abelian varieties are higher-dimensional generalizations of elliptic curves, defined as complete algebraic varieties that have a group structure. These varieties play a critical role in various areas of mathematics, including number theory and algebraic geometry, and they exhibit deep connections to concepts like complex multiplication, zeta functions, and modular forms.
Adelic points: Adelic points are a way to study rational points on algebraic varieties by considering them in a global context using the adeles, which combine local information from various places including the archimedean and non-archimedean completions of the number field. They allow mathematicians to understand solutions to polynomial equations in a more unified manner by incorporating both local and global perspectives, which is crucial for concepts like weak approximation and arithmetic equidistribution.
Affine varieties: Affine varieties are geometric objects that arise from solutions to systems of polynomial equations in an affine space. They can be understood as subsets of affine space that satisfy a specific set of polynomial equations, making them fundamental in algebraic geometry for studying shapes and their properties over a given field.
Alexander Grothendieck: Alexander Grothendieck was a groundbreaking mathematician known for his profound contributions to algebraic geometry, particularly in developing the modern foundations of the field. His work introduced key concepts such as schemes, sheaves, and cohomology, reshaping how mathematicians approach geometric problems and their algebraic underpinnings.
Bertini's Theorem: Bertini's Theorem is a fundamental result in algebraic geometry that asserts the general position of points in projective space, stating that a generic hyperplane section of a projective variety is smooth, provided that the variety itself is irreducible. This theorem has important implications for understanding the properties of varieties, particularly in weak approximation, arithmetic surfaces, and arithmetic threefolds, as it relates to the behavior of these structures under various conditions.
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.
Chevalley-Warning Theorem: The Chevalley-Warning Theorem is a result in algebraic geometry that provides conditions under which a system of polynomial equations has a solution in a given field. Specifically, it states that if the number of solutions to a system of polynomial equations is not too small, then there exists a solution in a certain number field. This theorem plays a significant role in understanding the structure of solutions to polynomial systems, particularly in relation to weak approximation.
Completeness: Completeness refers to a property of a mathematical space where every Cauchy sequence converges to a limit within that space. This concept is crucial for understanding various structures in mathematics, as it ensures that limits can be found without leaving the space. Completeness plays a vital role in local fields and weak approximation, as these areas rely on the ability to guarantee solutions exist within certain constraints.
Descent obstruction: Descent obstruction refers to the failure of a variety to have a rational point over a field due to certain obstructions that arise when considering its descent properties. These obstructions often relate to the behavior of rational points in relation to various cohomological theories, and they play a crucial role in understanding the solvability of equations over number fields. In particular, descent obstruction can be linked to other concepts such as the Brauer group and weak approximation, where it impacts the existence of rational solutions.
Diophantine equations: Diophantine equations are polynomial equations where the solutions are required to be integers or whole numbers. They are central to number theory and often relate to the search for rational points on algebraic varieties, connecting various mathematical concepts like algebraic geometry, arithmetic, and modular forms.
Function Fields: Function fields are fields consisting of rational functions, typically formed over a base field, which can be thought of as functions on algebraic varieties or schemes. They provide a framework for studying varieties over finite fields and are crucial in understanding various aspects of algebraic geometry and number theory.
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.
Henselian property: The henselian property is a condition in algebraic geometry that relates to the ability to lift solutions of polynomial equations modulo powers of a prime ideal. Specifically, it states that if a polynomial has a solution modulo a power of a prime, then it can be lifted to a solution in the local ring at that prime. This property is significant because it helps to understand how rational points can be approximated by solutions in a local setting, connecting the idea of local and global solutions.
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.
Lang's Theorem: Lang's Theorem is a fundamental result in arithmetic geometry that establishes the finiteness of rational points on certain algebraic varieties over number fields. This theorem provides important insight into how rational points behave in relation to various algebraic structures, particularly in the context of weak approximation and dynamical systems.
Linear algebraic groups: Linear algebraic groups are groups of matrices that are also algebraic varieties, meaning they can be described by polynomial equations. They play a crucial role in understanding symmetries in geometry and number theory, and they provide a framework for studying group actions on varieties. These groups can be classified into various types, such as unipotent, solvable, or reductive, each with unique properties and applications in different mathematical contexts.
Local fields: Local fields are a class of fields that are complete with respect to a discrete valuation and have finite residue fields. They play a crucial role in number theory and algebraic geometry, especially when examining properties of schemes over different completions, which allows for the study of rational points and the behavior of varieties over various bases. Their structure enables connections to Néron models, the Hasse principle, weak approximation, global class field theory, and p-adic numbers.
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.
Mori's Program: Mori's Program is a significant approach in algebraic geometry that aims to classify higher-dimensional varieties through the use of minimal models and the study of their birational properties. This program introduces the concept of 'polarization' to understand the geometric structure of varieties, while also addressing issues related to weak approximation, which deals with the ability to find rational points on varieties over local fields.
P-adic analysis: p-adic analysis is a branch of mathematics focused on the study of the p-adic numbers, which are a system of numbers that extend the rational numbers and provide a different way of measuring distances. This approach is particularly useful in number theory and algebraic geometry, allowing mathematicians to work with objects that are difficult to analyze using traditional methods, especially when looking at local properties of varieties over p-adic fields.
Patching Technique: The patching technique is a method used in arithmetic geometry to construct global objects from local data by gluing together information from various patches or local sections. This approach is crucial for addressing problems related to the existence of solutions over different fields, particularly in weak approximation scenarios where one seeks to ensure that local solutions can be extended to global ones.
Projective Varieties: Projective varieties are the sets of common solutions to homogeneous polynomial equations in projective space, which is a geometric structure that extends the notion of Euclidean space by adding 'points at infinity'. This concept plays a key role in algebraic geometry, allowing for the study of properties that remain invariant under projective transformations. Understanding projective varieties helps in solving equations, approximating solutions, and analyzing dynamical systems within projective spaces.
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.
Strong Approximation: Strong approximation refers to a concept in number theory and algebraic geometry that deals with the existence of solutions to equations over various fields, particularly concerning local and global fields. It indicates that if a certain condition holds locally for all completions of a field, then there is a solution in the global field as well. This idea connects closely with understanding rational points on algebraic varieties, examining the structure of rings of integers, and contrasts with weak approximation by emphasizing the robustness of solutions.
Valuation theory: Valuation theory is a mathematical framework that studies how to assign values to elements in a field, particularly focusing on places or valuations that capture the local properties of algebraic objects. It connects number theory, algebraic geometry, and topology by providing a way to measure the size and complexity of algebraic varieties and their rational points. This approach is essential in understanding weak approximation, as it allows for the comparison of local and global properties of solutions to equations.
Weak approximation: Weak approximation is a concept in number theory and algebraic geometry that refers to the ability to approximate rational points on varieties over global fields, such as the rational numbers or finite fields, using local information from completions at various places. This concept is significant as it relates to the existence of rational solutions and the distribution of these solutions across different local fields.
Zariski topology: Zariski topology is a type of topology used in algebraic geometry that defines the open sets of a space by using the vanishing sets of polynomials. It provides a way to study geometric objects algebraically, where closed sets are defined as the zero sets of collections of polynomials. This framework is crucial for understanding properties such as weak approximation, reciprocity laws in number theory, and the foundation of Grothendieck's more general topological concepts.