Zeta functions and Eisenstein series on classical groups

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.
• 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.