Eisenstein series and zeta functions on symplectic groups

1995 ◽  
Vol 119 (1) ◽  
pp. 539-584 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Eisenstein Series ◽  
Zeta Functions ◽  
Symplectic Groups

-ADIC -FUNCTIONS FOR ORDINARY FAMILIES ON SYMPLECTIC GROUPS

2019 ◽  
Vol 19 (4) ◽  
pp. 1287-1347 ◽  
Author(s):  
Zheng Liu
Keyword(s):  
Eisenstein Series ◽  
Symplectic Group ◽  
Symplectic Groups ◽  
Simple Strategy ◽  
Doubling Method

We construct the $p$-adic standard $L$-functions for ordinary families of Hecke eigensystems of the symplectic group $\operatorname{Sp}(2n)_{/\mathbb{Q}}$ using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group $\operatorname{Sp}(4n)_{/\mathbb{Q}}$, which guarantees the nonvanishing of local zeta integrals and allows us to $p$-adically interpolate the restrictions of the Siegel Eisenstein series to $\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$.

Poles ofL-functions and theta liftings for orthogonal groups

2009 ◽  
Vol 8 (4) ◽  
pp. 693-741 ◽  
Author(s):  
David Ginzburg ◽  
Dihua Jiang ◽  
David Soudry
Keyword(s):  
Eisenstein Series ◽  
Orthogonal Group ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Metaplectic Groups ◽  
Automorphic Representations ◽  
Domain Of Holomorphy ◽  
First Occurrence ◽  
Cuspidal Automorphic Representations

AbstractIn this paper, we prove that the first occurrence of global theta liftings from any orthogonal group to either symplectic groups or metaplectic groups can be characterized completely in terms of the location of poles of certain Eisenstein series. This extends the work of Kudla and Rallis and the work of Moeglin to all orthogonal groups. As applications, we obtain results about basic structures of cuspidal automorphic representations and the domain of holomorphy of twisted standardL-functions.

scholarly journals Hyperbolic-sine analogues of Eisenstein series, generalized Hurwitz numbers, and q-zeta functions

2014 ◽  
Vol 26 (4) ◽  
Author(s):  
Yasushi Komori ◽  
Kohji Matsumoto ◽  
Hirofumi Tsumura
Keyword(s):  
Eisenstein Series ◽  
Zeta Functions ◽  
Hurwitz Numbers ◽  
Hyperbolic Sine

scholarly journals Zeta functions on tori using contour integration

2015 ◽  
Vol 12 (02) ◽  
pp. 1550019
Author(s):  
Emilio Elizalde ◽  
Klaus Kirsten ◽  
Nicolas Robles ◽  
Floyd Williams
Keyword(s):  
Functional Equation ◽  
Eisenstein Series ◽  
Zeta Functions ◽  
Contour Integration ◽  
Alternative Proof ◽  
Functional Determinant ◽  
Dedekind Eta Function ◽  
Limit Formula ◽  
Complex Tori ◽  
The One

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla–Selberg series formula, for which an alternative proof is thereby given. In addition, a new proof of the functional determinant on the torus results, which does not use the Kronecker first limit formula nor the functional equation of the non-holomorphic Eisenstein series. As a bonus, several identities involving the Dedekind eta function are obtained as well.

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