13.3 Strong approximation theorem

The Strong Approximation Theorem bridges the gap between global and local properties in algebraic number fields. It shows that elements of a number field can approximate adeles and ideles, connecting finite and infinite places. This powerful result extends the Chinese Remainder Theorem to infinite sets of primes.

This theorem is crucial for understanding the distribution of prime ideals and solving Diophantine equations. It provides a framework for studying local-global principles in number theory, like the Hasse principle, and plays a key role in class field theory and the study of algebraic groups.

Strong Approximation Theorem

Fundamental Concepts and Definitions

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• Strong approximation theorem establishes relationship between global and local properties of algebraic number fields and their completions
• Adeles represent elements of the ring of adeles, forming restricted product of all completions of a number field with respect to its places
• Ideles constitute invertible elements of the ring of adeles, creating a group under multiplication
• Theorem states image of number field K under diagonal embedding becomes dense in ring of adeles A_K
• For idele group, theorem asserts image of K* (multiplicative group of K) becomes dense in idele class group C_K
• Two formulations exist for theorem addressing additive group of adeles and multiplicative group of ideles
• Generalizes Chinese Remainder Theorem to infinite sets of primes (Diophantine equations)

Mathematical Formulation and Implications

• For number field K and its ring of adeles A_K, diagonal embedding $\Delta: K \rightarrow A_K$ has dense image
• Formally expressed as $\overline{\Delta(K)} = A_K$ where overline denotes topological closure
• Idele formulation states $\overline{K^* \cdot \prod_v O_v^*} = I_K$ where $O_v^*$ represents units in local ring of integers
• Implies any adele $a \in A_K$ can be approximated arbitrarily closely by elements from K
• Allows approximation of elements in completions by elements from the original field K (p-adic numbers)
• Provides framework for studying distribution of prime ideals in number fields (Chebotarev density theorem)
• Facilitates investigation of local-global principles in number theory (Hasse principle)

Proving Strong Approximation

Foundational Concepts and Techniques

• Proof begins with ring of integers O_K of number field K and its completion at each prime ideal
• Construct ring of adeles A_K and group of ideles I_K using restricted products
• Apply weak approximation theorem stating K is dense in product of any finite set of its completions
• Utilize concept of fundamental domain for action of K* on A_K to simplify problem
• Employ finiteness of class number of K to show every idele class has representative with integral components almost everywhere
• Leverage properties of local fields and their completions to extend weak approximation to full adelic space
• Demonstrate any adele can be approximated arbitrarily closely by elements from diagonally embedded K

Key Steps in the Proof

• Start with arbitrary adele $a = (a_v) \in A_K$ and desired approximation accuracy $\epsilon > 0$
• Choose finite set S of places including all archimedean places and those where $|a_v|_v > 1$
• Apply weak approximation to find $x \in K$ close to $a_v$ for $v \in S$
• Utilize properties of restricted product to show $x$ is close to $a$ in adelic topology
• Employ ultrametric inequality in non-archimedean places to control approximation outside S
• Use strong triangle inequality to bound difference between $x$ and $a$ componentwise
• Conclude $x$ approximates $a$ within $\epsilon$ in adelic topology, proving density of K in A_K

Applications of Strong Approximation

Local-Global Principles and Equations

• Relate solutions of equations over local fields to solutions over global fields (Diophantine equations)
• Study distribution of prime ideals in number fields, particularly Dirichlet density
• Investigate existence of integral points on varieties defined over number fields (elliptic curves)
• Analyze structure of quadratic forms over number fields, relating to Hasse-Minkowski theorem
• Examine structure of algebraic groups over number fields and their local-global properties (Lie groups)
• Explore distribution of values of polynomials modulo prime ideals in number fields (Hilbert's irreducibility theorem)
• Construct elements in number fields with specified local properties (p-adic completions)

Advanced Number Theoretic Applications

• Apply theorem to study Galois groups of global fields and their completions
• Investigate ramification theory and its connection to local and global field extensions
• Analyze decomposition of prime ideals in extensions of number fields (splitting of primes)
• Study arithmetic properties of algebraic varieties over number fields using adelic methods
• Explore connections between strong approximation and zeta functions of number fields
• Investigate distribution of rational points on algebraic varieties using adelic techniques
• Apply theorem to study cohomology groups of adele rings and their arithmetic significance

Strong Approximation in Class Field Theory

Foundations and Connections

• Relate strong approximation theorem to existence and uniqueness of class fields for given modulus
• Study Galois group of maximal abelian extension of number field using theorem
• Investigate relationship between local and global class field theory through strong approximation
• Demonstrate theorem's contribution to proof of Artin reciprocity law, cornerstone of class field theory
• Explain role of strong approximation in formulation and proof of Chebotarev density theorem
• Examine structure of ray class groups and their relationship to abelian extensions of number fields
• Investigate theorem's contribution to understanding Brauer group of number field and its local-global properties

Advanced Topics in Class Field Theory

• Analyze idele class group and its role in formulation of class field theory using strong approximation
• Study norm residue symbol and its properties in context of strong approximation theorem
• Investigate connection between strong approximation and Hilbert's 12th problem (explicit class field theory)
• Examine role of strong approximation in proof of existence theorem for class fields
• Analyze decomposition and inertia groups in class field extensions using adelic techniques
• Study conductor-discriminant formula and its relation to strong approximation theorem
• Investigate connections between strong approximation and Langlands program in number theory