Open AccessZeta functions and Eisenstein series on classical groups
Author(s) -
Goro Shimura
Publication year - 1997
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.94.21.11133
Subject(s) - mathematics , eisenstein series , analytic continuation , pure mathematics , group (periodic table) , holomorphic function , quaternion , unitary state , l function , euler's formula , hermitian matrix , automorphic form , classical group , unitary group , algebra over a field , modular form , mathematical analysis , lie group , geometry , meromorphic function , chemistry , organic chemistry , political science , law
We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.