Local-Global Principle

Definition

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.

5 Must Know Facts For Your Next Test

1. The local-global principle helps determine the existence of rational points by checking local conditions at various places, particularly at the real numbers and p-adic numbers.
2. In the context of curves, the local-global principle often reveals that finding rational points can depend heavily on the nature of local fields, such as whether they are complete or discrete.
3. Weak approximation is a related concept where one seeks to find rational points that are close to given local points in a suitable sense, bridging local solutions with global ones.
4. The Artin reciprocity law is essential to understanding how fields relate through their Galois groups, helping elucidate connections between local and global fields.
5. Both global class field theory and local class field theory provide frameworks for understanding how abelian extensions behave locally and globally, which is crucial for applying the local-global principle.

Review Questions

• How does the local-global principle relate to finding rational points on algebraic varieties?
  • The local-global principle establishes that if a property holds locally at every place (like the existence of rational points), then it must also hold globally. This means that when trying to determine whether an algebraic variety has rational points, one can first check if there are solutions in all local fields (including p-adic and real fields). If it fails at any local level, then no global solution exists. Thus, understanding this relationship is key to solving equations over varieties.
• Discuss how the Hasse principle exemplifies the concept of the local-global principle in number theory.
  • The Hasse principle is a specific application of the local-global principle, focusing on polynomial equations over global fields. It asserts that a polynomial equation has a solution in a number field if and only if it has solutions in all completions of that field. This directly illustrates how checking local conditions at various primes allows mathematicians to infer information about global solutions, highlighting the importance of both local and global perspectives in number theory.
• Evaluate the impact of weak approximation on the local-global principle and its applications in arithmetic geometry.
  • Weak approximation enhances the local-global principle by providing conditions under which one can find rational points that are close to given local points. This concept allows for flexibility when dealing with varieties where exact solutions may not be easily computable. In arithmetic geometry, weak approximation demonstrates how even when rational points exist locally, they can be approximated globally within certain bounds. This further solidifies the connection between local properties and their implications for global structures.

Related terms

Local Fields: Local fields are complete fields with respect to a discrete valuation, typically used to study properties of numbers in a localized setting.

Rational Points: Rational points refer to points on algebraic varieties whose coordinates are rational numbers, crucial for understanding the solutions of polynomial equations.

Hasse Principle: The Hasse principle states that a polynomial equation has a solution over a global field if it has solutions in all local completions of that field.