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Euler products and Eisenstein series. (English) Zbl 0906.11020

Regional Conference Series in Mathematics. 93. Providence, RI: American Mathematical Society (AMS). xix, 259 p. (1997).
The main subject of this book is the study of the Euler product attached to a holomorphic cusp form on a hermitian symmetric domain which is an eigenfunction of an algebra of Hecke operators. Such a function can also be considered as a function on the group of adeles of a unitary group. Shimura’s treatment comprises all unitary groups belonging to a quadratic extension \(K/F\) where \(F\) is a totally real number field and \(K\) is totally imaginary, thus not only the quasi-split ones. This makes the analysis much more difficult than e.g. in the symplectic case.
Let \(f\) be a cusp form of the considered type on the group \(G\). It can be used to define an Eisenstein series on a bigger unitary group \(G_1\), and the product of this Eisenstein series with the Dirichlet series attached to \(f\) has an integral representation. The integral involves the pullback of an Eisenstein series on a quasi-split unitary group \(H = \text{U}(2n)\) in which \(G\times G_1\) can be embedded. In particular, there is an integral representation for the Dirichlet series (take \(G_{1} = \{1\}\)), but to obtain a general result a differential operator has to be applied to the Eisenstein series on \(H\). The analytic properties of the functions considered follow then from the analytic properties of Eisenstein series on quasi-split \(\text{U}(2n)\), which case was treated first.
In this work the local Euler factors are determined completely and explicitly, not only for the unitary case, but also for the symplectic and orthogonal cases. As an application of the method, a “mass formula” is proved for an anisotropic hermitian form. This is a generalized analogue of Siegel’s formula for the mass of a genus in the theory of quadratic forms, however Shimura’s formula gives the mass in question explicitly.
The book begins with an introduction to the general theory of automorphic forms on unitary, orthogonal and symplectic groups.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F55 Other groups and their modular and automorphic forms (several variables)
11M41 Other Dirichlet series and zeta functions
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