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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.


[Ba] L. Ballweg, “Die lokalen Cohomologiegruppen der Baily-Borel-Kompaktifizierung in generischen Randpunkten”, Dissertation University of Heidelberg, 1992
[Bo1] R. E. Borcherds, “Automorphic forms with singularities on Grassmannians”, Invent. Math 132 (1998), p. 491-562 Article |  MR 1625724 |  Zbl 0919.11036
[Bo2] R. E. Borcherds, “The Gross-Kohnen-Zagier theorem in higher dimensions”, Duke Math. J 97 (1999), p. 219-233 Article |  MR 1682249 |  Zbl 0967.11022
[Br1] J. H. Bruinier, “Borcherds products on $\Orth(2,l)$ and Chern classes of Heegner divisors”, Preprint, http://www.mathi.uni-heidelberg.de/~bruinier/, January 2000  Zbl 1004.11021
[Br2] J. H. Bruinier, “Borcherds products and Chern classes of Hirzebruch-Zagier divisors”, Invent. Math 138 (1999), p. 51-83 Article |  MR 1714336 |  Zbl 1011.11027
[Ca] H. Cartan, Fonctions automorphes, 1957/58
[EZ] M. Eichler & D. Zagier, The Theory of Jacobi Forms, Progress in Math 55, Birkhäuser, 1985  MR 781735 |  Zbl 0554.10018
[Fr1] E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, 1983  MR 871067 |  Zbl 0498.10016
[Fr2] E. Freitag, Hilbert Modular Forms, Springer-Verlag, 1990  MR 1050763 |  Zbl 0702.11029
[GrRe] H. Grauert & R. Remmert, “Plurisubharmonische Funktionen in komplexen Räumen”, Math. Zeitschr 65 (1956), p. 175-194 Article |  MR 81960 |  Zbl 0070.30403
[No] A. Nobs, “Die irreduziblen Darstellungen der Gruppen $SL_2({\Bbb Z}_p)$, insbesondere $SL_2({\Bbb Z}_2)$. I. Teil”, Comment Math. Helvetici 51 (1976), p. 465-489 Article |  MR 444787 |  Zbl 0346.20022
[Sh] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1971  MR 314766 |  Zbl 0221.10029
[Wal] J.-L. Waldspurger, “Engendrement par des séries thêta de certains espaces de formes modulaires”, Invent. Math. 50 (1979), p. 135-168 Article |  MR 517775 |  Zbl 0393.10025
[Wat] G. L. Watson, Integral quadratic forms, Cambridge University Press, 1960  MR 118704 |  Zbl 0090.03103