Mordell-Weil Group

Definition

The Mordell-Weil Group is an important concept in arithmetic geometry, specifically concerning the study of rational points on elliptic curves. It describes the group of rational points on an elliptic curve defined over a number field, equipped with a group structure based on the curve's addition law. Understanding the Mordell-Weil Group provides insights into the number of rational solutions and their algebraic properties, tying together various aspects of elliptic curves and number theory.

5 Must Know Facts For Your Next Test

1. The Mordell-Weil Theorem states that the Mordell-Weil Group of an elliptic curve over a number field is finitely generated, meaning it can be expressed as a direct sum of a free abelian group and a finite torsion subgroup.
2. The rank of the Mordell-Weil Group corresponds to the number of independent rational points on the elliptic curve, giving a measure of how many rational solutions exist.
3. Computing the Mordell-Weil Group often involves using techniques from both algebraic geometry and number theory, including descent methods and height functions.
4. Every elliptic curve defined over the rationals has a torsion subgroup that is well-understood, which plays a critical role in determining the structure of the Mordell-Weil Group.
5. The Mordell-Weil Group is closely related to various conjectures in number theory, including the Birch and Swinnerton-Dyer conjecture, which connects the rank of the group to the behavior of L-functions associated with the elliptic curve.

Review Questions

• Explain how the structure of the Mordell-Weil Group relates to rational points on an elliptic curve.
  • The structure of the Mordell-Weil Group directly reflects the rational points on an elliptic curve by forming a group under an addition operation defined by the curve's group law. Each rational point can be added together using this law, and through the Mordell-Weil Theorem, we learn that this group is finitely generated. The rank of this group provides essential information about how many independent rational points exist on the curve, bridging geometry and number theory.
• Discuss how computing the Mordell-Weil Group can utilize techniques from both algebraic geometry and number theory.
  • Computing the Mordell-Weil Group requires a blend of methods from algebraic geometry and number theory. Techniques such as descent methods help in identifying rational points through analysis of their local behavior at various primes. Height functions provide tools to measure sizes and help in understanding how many points are rational. By leveraging both fields, mathematicians can uncover deeper insights into the nature and structure of these groups.
• Evaluate the implications of the Mordell-Weil Theorem on our understanding of elliptic curves over number fields.
  • The Mordell-Weil Theorem profoundly impacts our understanding of elliptic curves over number fields by establishing that their Mordell-Weil Groups are finitely generated. This theorem indicates that while there may be infinitely many rational points, they can be categorized into a structured framework consisting of finitely many independent generators and a finite torsion subgroup. Such knowledge not only clarifies how we approach these curves but also leads to broader questions in arithmetic geometry, such as those posed by conjectures like Birch and Swinnerton-Dyer, connecting ranks to L-functions.

Related terms

Elliptic Curve:

A smooth, projective algebraic curve of genus one with a specified point, defined by a cubic equation in two variables.

Rational Points:

Points on an algebraic variety whose coordinates are rational numbers, often investigated in number theory.

Tate-Shafarevich Group: A group associated with an elliptic curve that measures the failure of the local-global principle for the curve's rational points.