Cm field

Definition

A cm field, or complex multiplication field, is a specific type of number field that arises in the context of abelian varieties and elliptic curves, characterized by having a non-trivial endomorphism ring that is isomorphic to an order in an imaginary quadratic field. These fields are important because they allow the construction of special points on elliptic curves and play a crucial role in the theory of complex multiplication, which connects number theory and algebraic geometry.

5 Must Know Facts For Your Next Test

1. Cm fields are related to the theory of complex multiplication, where elliptic curves have endomorphisms defined over these fields.
2. The endomorphism ring of an abelian variety over a cm field can give rise to interesting arithmetic properties, such as the existence of special values of L-functions.
3. An important example of a cm field is the field obtained from an imaginary quadratic number, which contains roots of unity.
4. The connection between cm fields and abelian varieties allows for explicit constructions of complex tori with rich structures.
5. Understanding cm fields is crucial for developing the theory of Shimura varieties and for studying rational points on abelian varieties.

Review Questions

• How does the concept of a cm field relate to the properties of elliptic curves?
  • Cm fields are directly linked to elliptic curves through their endomorphism rings. When an elliptic curve has complex multiplication, its endomorphism ring corresponds to a cm field, leading to unique properties such as special L-values and rich arithmetic structures. This relationship allows for the exploration of more advanced topics like modular forms and the Birch and Swinnerton-Dyer conjecture.
• Discuss the implications of having an endomorphism ring that is isomorphic to an order in an imaginary quadratic field for an abelian variety.
  • When an abelian variety has an endomorphism ring that is isomorphic to an order in an imaginary quadratic field, it indicates that the variety has additional symmetries and structure compared to general abelian varieties. This property can lead to specific arithmetic behavior, including relations to class groups and the behavior of points over finite fields. Such insights are valuable in number theory, particularly in understanding how these varieties behave under various morphisms.
• Evaluate the significance of cm fields in modern number theory and their impact on research areas such as arithmetic geometry.
  • Cm fields play a pivotal role in modern number theory by bridging concepts from algebraic geometry and arithmetic. Their importance is underscored by their connections to modular forms, Shimura varieties, and L-functions, which are central themes in current mathematical research. Understanding cm fields helps mathematicians tackle deep questions about rational points on varieties and offers insights into potential proofs for famous conjectures like the Langlands program. The intricate relationships they facilitate open up new avenues for exploration in both theoretical frameworks and practical applications.

Related terms

Abelian Variety: A higher-dimensional generalization of elliptic curves, which are complex tori that can be defined over any field and have a group structure.

Elliptic Curve: A smooth, projective algebraic curve of genus one with a specified point, often represented by an equation in the form $y^2 = x^3 + ax + b$.

Imaginary Quadratic Field: A number field of the form $ ext{Q}( ext{sqrt}-d)$, where $d$ is a positive integer, which has applications in complex multiplication and class field theory.