Abelian varieties

Definition

Abelian varieties are higher-dimensional generalizations of elliptic curves, defined as complete algebraic varieties that have a group structure. These varieties play a critical role in various areas of mathematics, including number theory and algebraic geometry, and they exhibit deep connections to concepts like complex multiplication, zeta functions, and modular forms.

5 Must Know Facts For Your Next Test

1. Abelian varieties can be defined over any field, but they are especially important over the fields of rational numbers and complex numbers due to their applications in number theory.
2. They possess a rich structure characterized by their dimension, which corresponds to the number of independent complex parameters needed to describe them.
3. Complex multiplication refers to the property of certain abelian varieties where the endomorphism ring is larger than just integers, allowing for deeper analysis in number theory.
4. The Hasse-Weil zeta function encodes important arithmetic information about abelian varieties and connects them with number theory through their points over finite fields.
5. Abelian varieties are central to many modern conjectures and results in arithmetic geometry, linking them to questions about rational points and modular forms.

Review Questions

• How do abelian varieties generalize the concept of elliptic curves, and what implications does this have for their study?
  • Abelian varieties extend the idea of elliptic curves to higher dimensions, allowing for more complex structures and interactions. This generalization means that many results known for elliptic curves can be adapted to higher-dimensional cases, leading to new insights in areas such as number theory and algebraic geometry. Additionally, studying these varieties enriches our understanding of rational points and their distributions over various fields.
• Discuss the significance of complex multiplication in relation to abelian varieties and its impact on modern number theory.
  • Complex multiplication is crucial for certain abelian varieties as it provides a richer structure than what is found in ordinary cases. It allows mathematicians to study these varieties through their endomorphisms, leading to deeper connections with modular forms and L-functions. This area has seen significant development in modern number theory, particularly in efforts to understand rational points and prove conjectures like those related to the Birch and Swinnerton-Dyer conjecture.
• Evaluate the role of abelian varieties in formulating Serre's modularity conjecture and Fontaine-Mazur conjecture within arithmetic geometry.
  • Abelian varieties serve as a foundational element in both Serre's modularity conjecture and Fontaine-Mazur conjecture, highlighting their importance in connecting different areas within arithmetic geometry. Serre's conjecture proposed a link between Galois representations and modular forms through abelian varieties, while Fontaine-Mazur's conjecture addresses the interplay between p-adic representations and their moduli. Understanding abelian varieties is essential for exploring these conjectures, as they bridge classical number theory with modern geometric methods.

Related terms

Elliptic Curves: A special case of abelian varieties of dimension one, elliptic curves are smooth projective curves of genus one with a specified point that serve as the identity element for the group structure.

Torsion Points: Points on an abelian variety that have finite order under the group operation, meaning there exists a positive integer such that when added to itself that many times yields the identity element.

Theorems of Mordell-Weil: A set of results concerning the group of rational points on abelian varieties, establishing that this group is finitely generated, providing key insights into their structure.