Theta Functions : From the Classical to the Modern

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Title page
Contents
Preface
Ramanujan's Theory of Theta-Functions, Bruce C. Berndt
	1. Definitions
	2. Ramanujan's ₁?₁ summation and the Jacobi triple product identity
	3. Basic multiplicative and additiveproperties, including Schröter's formulas
	4. Quintuple product identity
	5. Applications to Lambert series and sums of squares
	6. Elliptic integrals
	7. Inversion formulas
	8. Catalogue of theta-function evaluations
	9. Modular equations
	10. Eta-function identities
	11. Theta-function identities and work of R. J. Evans
	12. Ramanujan's Eisenstein series P, Q, and R
	13. Ramanujan's theories of elliptic functions to alternative bases
	References
Eisenstein Series and Theta Functions on the Metaplectic Group, Jeff Hoffstein
	Introduction
	1. Eisenstein series and theta functions on the double cover of GL(2)
	2. The n-fold cover of GL(2)
	3. Eisenstein series and theta functions over function fields
	4. Some GL(3) examples
	5. Theta functions on the n-fold cover of GL(r)
	6. The Group GSp(2n)
	References
Weil Representation, Howe Duality, and the Theta Correspondence, Dipendra Prasad
	1. Heisenberg group
	2. Metaplectic group and the Weil representation
	3. Dual reductive pairs
	4. Howe duality
	5. Howe conjecture in the Archimedean case
	6. The spherical case
	7. Seesaw pairs
	8. The theta correspondence
	9. Questions
	References
On Theta-Series Liftings for Unitary Groups, Stephen Gelbart
	Introductory remarks
	1. Weil's representation and theta-series
	2. Howe's correspondence and theta-series liftings
	3. Specialization to the unitary group U(3)
	4. Trace formula results
	5. L-functions for U(3)
	6. Characterization of endoscopic representations
	Appendix. Explicit formulas for the Weil representation restricted to unipotent elements of U(V) X U(Φ')
	References
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