Weil Pairing

Definition

Weil pairing is a bilinear pairing defined between the points of an algebraic variety and the elements of its Jacobian, which captures important geometric and arithmetic information about the variety. This concept is crucial for understanding how torsion points on abelian varieties interact with each other and helps establish connections with reciprocity laws, complex tori, and periodic points in the study of arithmetic geometry.

5 Must Know Facts For Your Next Test

1. The Weil pairing is often denoted as $e(P, Q)$ for two points $P$ and $Q$ on an abelian variety, producing an element in the finite field that relates to their respective orders.
2. It can be used to show whether two torsion points are linearly independent or not, which is significant in studying their properties.
3. The Weil pairing satisfies certain properties, including alternatingness, bilinearity, and non-degeneracy, which make it useful for constructing isogenies between abelian varieties.
4. In the context of complex tori, Weil pairings help understand the complex structure and its relation to algebraic properties.
5. The pairing plays a critical role in explicit reciprocity laws, linking local data from abelian varieties to global properties and behaviors.

Review Questions

• How does the Weil pairing demonstrate relationships among torsion points on an abelian variety?
  • The Weil pairing provides a way to measure the interaction between torsion points on an abelian variety. By calculating the Weil pairing for two torsion points, you can determine if they are linearly independent. If the Weil pairing yields a non-zero value, it indicates that these two points do not lie in a common subgroup generated by other points. This insight helps classify and understand the structure of torsion subgroups within abelian varieties.
• In what ways does the Weil pairing contribute to explicit reciprocity laws in number theory?
  • The Weil pairing connects local conditions with global properties through its bilinear structure. In explicit reciprocity laws, it serves as a bridge between the arithmetic of torsion points and their behavior under various mappings. By understanding how these pairings function under different operations, mathematicians can derive results about field extensions and solve problems related to rational points on varieties. This connection enhances our grasp of the underlying arithmetic structure tied to these varieties.
• Evaluate the significance of Weil pairings in understanding complex tori and their geometric interpretations.
  • Weil pairings play a pivotal role in linking the arithmetic properties of complex tori with their geometric interpretations. Through this pairing, one can examine how complex structures interact with algebraic ones, revealing insights into symmetry and duality within these tori. For instance, when analyzing complex tori as quotient spaces of $ ext{C}^g$ by lattices, Weil pairings help explain how periods relate to divisor classes. This understanding is essential for unraveling deeper relationships between geometry and number theory.

Related terms

Abelian Variety: An abelian variety is a complete algebraic variety that has a group structure, allowing for the addition of points. They are fundamental objects in algebraic geometry and have applications in number theory.

Jacobian: The Jacobian of a curve is an abelian variety that parametrizes line bundles on the curve. It plays a key role in the study of divisors and linear equivalence.

Torsion Point: A torsion point is a point on an abelian variety that has a finite order with respect to the group operation. This means that there exists some positive integer such that adding the point to itself that many times results in the identity element.