Cm points

Definition

CM points, or complex multiplication points, are specific points in the moduli space of abelian varieties that exhibit complex multiplication by an order in a number field. These points play a crucial role in the theory of abelian varieties, allowing for a deeper understanding of their structure and properties, particularly in connection with elliptic curves and their modular forms. CM points are linked to special endomorphisms, leading to rich arithmetic structures and connections to algebraic geometry.

5 Must Know Facts For Your Next Test

1. CM points correspond to abelian varieties that have extra symmetries due to their endomorphisms being defined over a number field.
2. The existence of CM points allows for special relationships between the geometry of these varieties and the arithmetic of number fields.
3. CM points often arise in the study of elliptic curves, especially those defined over number fields that have complex multiplication.
4. The theory surrounding CM points has important implications for number theory, particularly in the construction of L-functions and understanding their properties.
5. CM points can be used to construct special types of modular forms, which relate to deep areas such as the Langlands program.

Review Questions

• How do CM points enhance our understanding of the structure of abelian varieties?
  • CM points enhance our understanding by providing insights into the endomorphisms of abelian varieties, which are special homomorphisms defined over number fields. These endomorphisms introduce symmetries that allow mathematicians to classify and understand different forms of abelian varieties. The existence of these points connects geometry with number theory, offering a richer framework for exploring properties like reduction modulo primes.
• Discuss the relationship between CM points and elliptic curves, particularly in terms of their modularity.
  • CM points are intimately connected with elliptic curves that exhibit complex multiplication, which means their endomorphism rings have additional structure. This relationship leads to results in modularity; specifically, these curves can often be associated with specific modular forms. The modularity theorem highlights how every elliptic curve over the rationals can be associated with a modular form, and CM points serve as critical examples that illustrate this connection.
• Evaluate how the study of CM points influences modern research areas like the Langlands program.
  • The study of CM points significantly influences modern research areas such as the Langlands program by bridging gaps between number theory and representation theory. CM points help to construct L-functions associated with abelian varieties, which are vital in connecting different areas within mathematics. By understanding the role of these points, researchers can gain insights into deeper conjectures about automorphic forms and their relationships with Galois representations, making them essential in contemporary mathematical discourse.

Related terms

Abelian Variety: A projective algebraic variety that is also a group variety, meaning it has a group structure that is compatible with its algebraic structure.

Endomorphism: A homomorphism from an object to itself; in the context of abelian varieties, endomorphisms can provide additional structure and symmetry.

Moduli Space: A geometric space that classifies objects of a certain type up to isomorphism; for abelian varieties, it encodes their various forms and structures.