11.3 Local-global principle

The local-global principle 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, quadratic forms, 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 global field can be determined by examining properties over all completions of the field
• Global field typically refers to a number field in algebraic number theory
• Completions include p-adic fields 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 local solvability does not imply global solvability

Applications in algebraic number theory

• Determines existence of solutions to Diophantine equations
• Studies properties of quadratic forms
• Analyzes rational points on algebraic varieties (elliptic curves)
• Investigates local-global principles for more general algebraic structures (group cohomology)
• Examines Galois cohomology and its relationship to local-global phenomena
• Explores connections with class field theory and adelic methods

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 Hensel's lemma 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 Chinese Remainder Theorem to combine local solutions
• Apply Hasse-Minkowski theorem for quadratic forms
• Analyze elliptic curves using local-global methods (Mordell-Weil theorem)
• Study higher degree equations (cubic surfaces, equations of degree ≥ 4)
• Implement computational methods to check local solvability efficiently
• Explore connections with modular forms and Galois representations (Serre's modularity conjecture)

The Hasse principle

Quadratic forms and the Hasse-Minkowski theorem

• Hasse principle 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 Tate-Shafarevich group
• Determine rank of elliptic curves 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 Selmer groups and their role in understanding the failure of the Hasse principle
• Study height functions 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: $3x^3 + 4y^3 + 5z^3 = 0$
• Birch and Swinnerton-Dyer's 1975 counterexample: $2y^2 = 1 - 17x^4$
• Cassels and Guy's 1966 diagonal cubic surface: $3x^3 + 4y^3 + 5z^3 = 0$
• 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 Brauer-Manin obstruction often relates to failure of Hasse principle
• Étale-Brauer Hasse principle refines original principle to account for some failures
• Birch and Swinnerton-Dyer conjecture connects to failures of Hasse principle for elliptic curves
• Develop new tools in arithmetic geometry to study rational points on varieties
• Investigate transcendental methods (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