Beilinson Conjectures

Definition

The Beilinson Conjectures are a set of conjectures in algebraic geometry and number theory that link the ranks of certain cohomology groups of algebraic varieties with special values of L-functions associated with those varieties. These conjectures propose a profound relationship between geometry, arithmetic, and the behavior of L-functions, which play a central role in modern number theory.

5 Must Know Facts For Your Next Test

1. The Beilinson Conjectures were proposed by Alexander Beilinson in the 1980s as part of his work on motives and their relationship to L-functions.
2. These conjectures suggest that for a smooth projective variety, the rank of its Chow group is closely linked to the order of vanishing of its associated L-function at specific points.
3. The conjectures unify various results from algebraic geometry and number theory, establishing a bridge between them that has led to further research in both areas.
4. The Beilinson Conjectures have implications for understanding the Birch and Swinnerton-Dyer Conjecture, particularly in their treatment of rational points on elliptic curves.
5. These conjectures remain unproven in general, but they have been verified in several special cases, contributing to the ongoing development of ideas in motivic cohomology.

Review Questions

• How do the Beilinson Conjectures connect algebraic geometry and number theory?
  • The Beilinson Conjectures establish a connection between algebraic geometry and number theory by linking the ranks of cohomology groups of algebraic varieties with special values of their associated L-functions. This relationship indicates that geometric properties can influence arithmetic behavior, demonstrating how methods from both fields can be applied to gain insights into each other. The conjectures suggest that understanding one aspect can lead to understanding the other, paving the way for deeper investigations in both domains.
• Discuss the significance of L-functions in relation to the Beilinson Conjectures.
  • L-functions play a crucial role in the Beilinson Conjectures as they provide the analytic tools needed to study the arithmetic properties of algebraic varieties. According to these conjectures, there is a specific relationship between the ranks of Chow groups and the behavior of these L-functions at critical points. This interaction underlines the importance of L-functions in linking geometry with number theory, as they encapsulate vital information about the variety's rational points and cohomological properties.
• Evaluate the implications of proving or disproving the Beilinson Conjectures on modern mathematics.
  • Proving or disproving the Beilinson Conjectures would have significant implications for modern mathematics, particularly in our understanding of the interplay between algebraic geometry and number theory. A proof could validate many existing results and open new pathways for research, impacting related conjectures like the Birch and Swinnerton-Dyer Conjecture. On the other hand, a disproof might require mathematicians to rethink current theories about motives and L-functions, prompting an investigation into alternative frameworks or methods that better describe these relationships.

Related terms

Cohomology:

A mathematical tool used to study the properties of topological spaces through algebraic means, often used in conjunction with sheaf theory and homological algebra.

L-function: A complex function associated with a number-theoretic object (like an algebraic variety) that encodes significant arithmetic information and generalizes the Riemann zeta function.

Algebraic Geometry:

A branch of mathematics that studies geometric properties of solutions to polynomial equations, bridging algebra and geometry.