## Theta Functions: From the Classical to the ModernThis book contains lectures on theta functions written by experts well known for excellence in exposition. The lectures represent the content of four courses given at the Centre de Recherches Mathematiques in Montreal during the academic year 1991-1992, which was devoted to the study of automorphic forms. Aimed at graduate students, the book synthesizes the classical and modern points of view in theta functions, concentrating on connections to number theory and representation theory. An excellent introduction to this important subject of current research, this book is suitable as a text in advanced graduate courses. |

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### Contents

Ramanujans Theory of ThetaFunctions | 1 |

Eisenstein Series and Theta Functions on the Metaplectic Group | 65 |

Weil Representation Howe Duality and the Theta Correspondence | 105 |

On ThetaSeries Liftings for Unitary Groups | 129 |

### Common terms and phrases

analogous automorphic forms automorphic representation classical complete the proof conjecture cover of GL(2 cubic cusp forms cuspidal representation defined definition denote Dirichlet series discrete double cover dual pair dual reductive pair Eisenstein series elliptic endoscopic equation of degree Euler product example finite formula Fourier coefficients function field functional equation functoriality Gauss sums GL(n GL(r global Heisenberg group homomorphism integral irreducible representation L-functions L-packet L-series Langlands Lemma Math metaplectic covering metaplectic group metaplectic representation modular equations modular forms Mp(W multiplicity n-cover n-fold cover non-metaplectic form non-trivial non-zero notation Note Number Theory parabolic Eisenstein series Piatetski-Shapiro pole prime product identity PROPOSITION prove quadratic theta function Rankin-Selberg convolution REMARK representation of G representation of O(V representation of Sp(W representation TT residue restricted Section Shimura correspondent square free Suppose symplectic group Theorem theta function theta-functions theta-lift theta-series theta-series liftings unique unitary group unramified Waldspurger Weil representation zero