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PERIODS OF EISENSTEIN AND POINCARÉ SERIES

Leon Ehrenpreis

in The Universality of the Radon Transform

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198509783
eISBN:
9780191709166
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509783.003.0010
Subject:
Mathematics, Mathematical Physics

This chapter shows how the book's main theory applies to computing the periods of Eisenstein and Poincaré series. Our Poincaré series are eigenfunctions of the Laplacian, and they arise naturally in ... More


Whittaker Functions

Ben Brubaker, Daniel Bump, and Solomon Friedberg

in Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory (AM-175)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691150659
eISBN:
9781400838998
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691150659.003.0004
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter shows that Weyl group multiple Dirichlet series are expected to be Whittaker coefficients of metaplectic Eisenstein series. The fact that Whittaker coefficients of Eisenstein series ... More


Eisenstein Cohomology

Günter Harder and A. Raghuram

in Eisenstein Cohomology for GL and the Special Values of Rankin-Selberg L-Functions: (AMS-203)

Published in print:
2019
Published Online:
September 2020
ISBN:
9780691197890
eISBN:
9780691197937
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691197890.003.0006
Subject:
Mathematics, Number Theory

This chapter provides the Eisenstein cohomology. It begins with the Poincaré duality and maximal isotropic subspace of boundary cohomology. Here, the chapter considers the compatibility of duality ... More


Tokuyama’s Theorem

Ben Brubaker, Daniel Bump, and Solomon Friedberg

in Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory (AM-175)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691150659
eISBN:
9781400838998
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691150659.003.0005
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces the Tokuyama's Theorem, first by writing the Weyl character formula and restating Schur polynomials, the values of the Whittaker function multiplied by the normalization ... More


Trace of the Generating Series

Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang

in The Gross-Zagier Formula on Shimura Curves

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691155913
eISBN:
9781400845644
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691155913.003.0004
Subject:
Mathematics, Number Theory

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the ... More


Weil Representation and Waldspurger Formula

Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang

in The Gross-Zagier Formula on Shimura Curves

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691155913
eISBN:
9781400845644
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691155913.003.0002
Subject:
Mathematics, Number Theory

This chapter reviews some basic results on Weil representations, theta liftings and Eisenstein series. In particular, it introduces a proof of the Waldspurger formula. The theory of Weil ... More


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