Abelian surfaces with CM

Definition

Abelian surfaces with complex multiplication (CM) are two-dimensional algebraic varieties that exhibit special endomorphisms corresponding to certain algebraic integers. They arise in the context of complex geometry and number theory, where they provide a rich interplay between algebraic structures and geometric properties. These surfaces have applications in various areas such as arithmetic geometry, particularly in understanding the arithmetic of abelian varieties and their endomorphisms.

5 Must Know Facts For Your Next Test

1. Abelian surfaces with CM have endomorphism rings that are often isomorphic to orders in imaginary quadratic fields, which means their geometric properties are deeply tied to number theory.
2. These surfaces exhibit unique lattice structures that make them interesting objects of study in both algebraic geometry and arithmetic geometry.
3. The theory of complex multiplication allows for the construction of special points on these surfaces, leading to important insights into their moduli spaces.
4. The existence of CM provides a means to define and study automorphic forms and L-functions, which are central themes in modern number theory.
5. Abelian surfaces with CM can be used to construct explicit examples of abelian varieties that exhibit particular behaviors under morphisms and mappings.

Review Questions

• How does the presence of complex multiplication influence the structure of abelian surfaces?
  • Complex multiplication introduces a richer structure to abelian surfaces by allowing for additional endomorphisms beyond simple integer multiplication. This results in the endomorphism ring having interesting algebraic properties, often linked to orders in imaginary quadratic fields. Consequently, this influences their geometric properties and leads to significant connections with number theory, particularly in studying rational points on these surfaces.
• Discuss the implications of the endomorphism ring being isomorphic to an order in an imaginary quadratic field for abelian surfaces with CM.
  • When the endomorphism ring of an abelian surface with CM is isomorphic to an order in an imaginary quadratic field, it implies that there is a deep connection between the geometry of the surface and number theoretic properties. This connection allows for the application of algebraic number theory techniques to understand the arithmetic behavior of the surface. It also facilitates the classification of such surfaces and provides a framework for studying their moduli spaces.
• Evaluate how abelian surfaces with CM contribute to modern developments in arithmetic geometry and number theory.
  • Abelian surfaces with CM play a pivotal role in modern arithmetic geometry by serving as examples that bridge complex geometry with number theory. Their unique structures allow mathematicians to explore new avenues in understanding modular forms, L-functions, and rational points on varieties. The interaction between these surfaces and concepts from algebraic topology also contributes to significant advances in understanding elliptic curves and higher-dimensional varieties, making them essential in the ongoing research landscape.

Related terms

Complex Multiplication (CM): A phenomenon where an abelian variety has endomorphisms that are not just given by multiplication by integers, but also include certain non-integer algebraic elements.

Endomorphism Ring: The set of endomorphisms of an abelian variety forms a ring, which can provide insights into its structure and properties.

Tate Module: A construction associated with abelian varieties that encodes information about their p-torsion points, helping to study their properties over finite fields.