Complex multiplication

Definition

Complex multiplication refers to a special technique in number theory involving the multiplication of complex numbers, particularly in relation to elliptic curves and their endomorphisms. This concept connects the theory of elliptic curves with algebraic number theory, offering insights into the structure of class fields and abelian varieties, which are essential for understanding certain arithmetic properties in number fields.

5 Must Know Facts For Your Next Test

1. Complex multiplication allows for the explicit construction of abelian extensions of number fields by utilizing the endomorphisms of elliptic curves.
2. The theory relies heavily on the relationships between complex numbers and modular forms, leading to significant results like the L-functions associated with elliptic curves.
3. One crucial application is the explicit computation of class numbers in imaginary quadratic fields, which informs us about the distribution of prime ideals.
4. In complex multiplication, specific lattices are used to define elliptic curves, showcasing how algebraic structures can influence number theoretic properties.
5. This technique provides powerful methods to construct class fields explicitly, revealing deeper connections between different areas of mathematics.

Review Questions

• How does complex multiplication relate to elliptic curves and their endomorphisms?
  • Complex multiplication is integral to understanding elliptic curves because it examines how these curves can be manipulated through their endomorphisms. Endomorphisms are functions that map an elliptic curve to itself, and complex multiplication specifically studies cases where these functions are defined over complex numbers. This relationship allows mathematicians to explore deeper properties of elliptic curves and their connections to various number fields.
• What role does complex multiplication play in class field theory and its applications?
  • In class field theory, complex multiplication serves as a bridge between abelian extensions of number fields and the structure of elliptic curves. By using the endomorphisms provided by complex multiplication, one can construct class fields explicitly. This construction helps solve various problems in number theory, such as calculating class numbers and understanding the behavior of primes in different number fields.
• Evaluate the impact of complex multiplication on the arithmetic properties of imaginary quadratic fields.
  • Complex multiplication significantly impacts the arithmetic properties of imaginary quadratic fields by allowing for explicit computations of class numbers and revealing patterns in prime ideal distributions. Through its connection with elliptic curves, it provides tools for classifying abelian extensions and understanding Galois representations. This evaluation not only enhances our knowledge of these number fields but also illustrates how deeply interconnected various mathematical concepts can be within number theory.

Related terms

Elliptic Curve: A smooth, projective algebraic curve of genus one, equipped with a specified point, which plays a significant role in number theory and cryptography.

Endomorphism: A morphism from a mathematical object to itself, particularly relevant in the context of elliptic curves and their structure.

Class Field Theory:

A major branch of algebraic number theory that studies abelian extensions of number fields and connects them to the ideal class groups.