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Jan Hendrik Bruinier; Eberhard Freitag
Local Borcherds products
(Produits de Borcherds locaux)
Annales de l'institut Fourier, 51 no. 1 (2001), p. 1-26, doi: 10.5802/aif.1812
Article PDF | Reviews Zbl 0966.11021
Class. Math.: 11F55, 14L35
Keywords: automorphic forms, automorphic product, orthogonal group, Heegner divisor, local Picard group

Résumé - Abstract

The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a Hermitean symmetric quotient of type ${\rm O}(2,n)$ is computed. The main ingredient is a local version of Borcherds' automorphic products. The local obstructions for a Heegner divisor to be principal are given by certain theta series with harmonic coefficients. Sometimes they generate Borcherds' space of global obstructions. In these particular cases we obtain a simple proof of a result due to the first author: Suppose that $\Gamma\subset{\rm O}(2,n)$ is the orthogonal group attached to an even unimodular lattice. Then every meromorphic modular form for $\Gamma$, whose zeros and poles lie on Heegner divisors, is given by a Borcherds product.

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