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Related Experiment Videos

Zeta functions and Eisenstein series on classical groups.

G Shimura1

  • 1Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ 08544, USA.

Proceedings of the National Academy of Sciences of the United States of America
|October 19, 2001
PubMed
Summary

Researchers constructed an Euler product from automorphic form Hecke eigenvalues on unitary groups over CM fields. This Euler product and related Eisenstein series were proven to have analytic continuation, yielding a class number formula.

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Area of Science:

  • Number Theory
  • Automorphic Forms
  • Representation Theory

Background:

  • Automorphic forms and their associated L-functions are central objects in number theory.
  • Euler products encode arithmetic information of automorphic forms.
  • Analytic continuation of these objects is crucial for understanding their properties.

Purpose of the Study:

  • To construct an Euler product from Hecke eigenvalues of holomorphic automorphic forms on unitary groups over CM fields.
  • To prove the analytic continuation of this Euler product to the entire complex plane.
  • To establish an explicit class number formula for totally definite hermitian forms over CM fields.

Main Methods:

  • Construction of an Euler product using Hecke eigenvalues.
  • Techniques for proving analytic continuation of automorphic L-functions.

Related Experiment Videos

  • Analysis of Eisenstein series on unitary groups.
  • Application of automorphic methods to number theoretic problems.

    • Main Results:

    • The Euler product associated with a holomorphic automorphic form on a unitary group over a CM field admits analytic continuation.
    • The analytic continuation of a specific Eisenstein series on a related unitary group is established.
    • An explicit formula for the class number of a totally definite hermitian form over a CM field is derived.

    Conclusions:

    • The methods provide a powerful framework for studying automorphic forms and their L-functions.
    • The results connect deep concepts in number theory, including automorphic forms and class numbers.
    • This work offers new tools and insights into the arithmetic of unitary groups and hermitian forms.