Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors

Jan H. Bruinier3540433201, 9783540433200

Around 1994 R. Borcherds discovered a new type of meromorphic modular form on the orthogonal group $O(2,n)$. These “Borcherds products” have infinite product expansions analogous to the Dedekind eta-function. They arise as multiplicative liftings of elliptic modular forms on $(SL)_2(R)$. The fact that the zeros and poles of Borcherds products are explicitly given in terms of Heegner divisors makes them interesting for geometric and arithmetic applications. In the present text the Borcherds’ construction is extended to Maass wave forms and is used to study the Chern classes of Heegner divisors. A converse theorem for the lifting is proved.

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Table of contents :
front-matter……Page 1
1.1 The Weil representation……Page 9
1.2.1 Poincaré series……Page 13
1.2.2 The Petersson scalar product……Page 16
1.2.3 Eisenstein series……Page 17
1.3 Non-holomorphic Poincar´e series of negative weight……Page 21
1.1 The Weil representation……Page 33
1.2.1 Poincaré series……Page 37
1.2.2 The Petersson scalar product……Page 40
1.2.3 Eisenstein series……Page 41
1.3 Non-holomorphic Poincar´e series of negative weight……Page 45
2.1 Siegel theta functions……Page 57
2.2 The theta integral……Page 64
2.3 Unfolding against $F_{beta,m}$……Page 72
2.4 Unfolding against $Theta_L$……Page 75
3.1 Lorentzian lattices……Page 80
3.1.1 The hyperbolic Laplacian……Page 89
3.2 Lattices of signature $(2, l)$……Page 90
3.3 Modular forms on orthogonal groups……Page 101
3.4 Borcherds products……Page 104
3.4.1 Examples……Page 108
4.1 The invariant Laplacian……Page 112
4.2 Reduction theory and $L^p$-estimates……Page 120
4.3 Modular forms whose zeros and poles lie on Heegner divisors……Page 129
5. Chern classes of Heegner divisors……Page 136
5.1 A lifting into the cohomology……Page 142
5.2 Modular forms whose zeros and poles lie on Heegner divisors II……Page 154
References……Page 158
Subject Index and Notation Index……Page 162