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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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