Hilbert Modular Forms

Definition

Hilbert modular forms are a generalization of classical modular forms that arise in the study of abelian varieties over totally real fields. These forms are functions that are holomorphic on the upper half-space and satisfy specific transformation properties under the action of Hilbert modular groups, connecting number theory with geometry. They play an important role in various areas, including the study of L-functions and the arithmetic of algebraic varieties.

5 Must Know Facts For Your Next Test

1. Hilbert modular forms can be thought of as functions defined on the product of upper half-planes corresponding to real embeddings of a number field.
2. They are crucial for constructing L-functions, which encode important number-theoretic information and have deep connections to various areas in mathematics.
3. The space of Hilbert modular forms has a rich structure, with a correspondence between these forms and certain algebraic objects like abelian varieties.
4. Just like classical modular forms, Hilbert modular forms can be decomposed into eigenforms, leading to insights into their arithmetic properties.
5. Hilbert modular forms can also be classified according to their level, weight, and character, leading to different types of forms that arise in various contexts.

Review Questions

• How do Hilbert modular forms generalize classical modular forms, and what implications does this have for their study?
  • Hilbert modular forms extend the concept of classical modular forms by incorporating functions defined over totally real fields rather than just over the rationals. This generalization allows for a richer interaction with number theory and algebraic geometry, as it connects various structures such as abelian varieties to L-functions. The study of these forms reveals deeper insights into the arithmetic properties of more complex number fields.
• Discuss the relationship between cusp forms and Hilbert modular forms, focusing on their significance within number theory.
  • Cusp forms are a subset of modular forms that vanish at all cusps and play a significant role in both classical and Hilbert modular form theories. In the context of Hilbert modular forms, cusp forms help in understanding the geometric aspects related to divisors on abelian varieties and lead to important results concerning L-functions. The interplay between these two types of forms enhances our understanding of how symmetry in functions relates to deep arithmetic structures.
• Evaluate how newforms contribute to the theory of Hilbert modular forms and their applications in modern number theory.
  • Newforms serve as a bridge between different levels of Hilbert modular forms, capturing essential information about the structure and behavior of these forms at various levels. Their study can reveal intricate relationships between algebraic objects like Galois representations and L-functions. By analyzing newforms within the context of Hilbert modular forms, mathematicians can gain insights into more complex problems in number theory, contributing to advancements in understanding arithmetic geometry and its applications.

Related terms

Modular Forms: Functions that are analytic and exhibit certain symmetry properties under the action of a modular group, used in various areas such as number theory and algebraic geometry.

Cusp Forms: A special type of modular form that vanishes at all cusps of the modular curve, representing a more refined structure within the theory of modular forms.

Newforms:

A particular kind of modular form that is associated with a new level structure and captures essential arithmetic information about modular forms.