Local Borcherds products

Jan Hendrik Bruinier[1]; Eberhard Freitag[1]

  • [1] Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg (Allemagne)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 1, page 1-26
  • ISSN: 0373-0956

Abstract

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The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a Hermitean symmetric quotient of type 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 Γ ⊂ O ( 2 , n ) is the orthogonal group attached to an even unimodular lattice. Then every meromorphic modular form for Γ , whose zeros and poles lie on Heegner divisors, is given by a Borcherds product.

How to cite

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Bruinier, Jan Hendrik, and Freitag, Eberhard. "Local Borcherds products." Annales de l’institut Fourier 51.1 (2001): 1-26. <http://eudml.org/doc/115908>.

@article{Bruinier2001,
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.},
affiliation = {Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg (Allemagne); Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg (Allemagne)},
author = {Bruinier, Jan Hendrik, Freitag, Eberhard},
journal = {Annales de l’institut Fourier},
keywords = {automorphic forms; automorphic product; orthogonal group; Heegner divisor; local Picard group; Automorphic forms; automorphic products; orthogonal groups; Heegner divisors; local Picard groups},
language = {eng},
number = {1},
pages = {1-26},
publisher = {Association des Annales de l'Institut Fourier},
title = {Local Borcherds products},
url = {http://eudml.org/doc/115908},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Bruinier, Jan Hendrik
AU - Freitag, Eberhard
TI - Local Borcherds products
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 1
SP - 1
EP - 26
AB - 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.
LA - eng
KW - automorphic forms; automorphic product; orthogonal group; Heegner divisor; local Picard group; Automorphic forms; automorphic products; orthogonal groups; Heegner divisors; local Picard groups
UR - http://eudml.org/doc/115908
ER -

References

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  1. L. Ballweg, Die lokalen Cohomologiegruppen der Baily-Borel-Kompaktifizierung in generischen Randpunkten, (1992) 
  2. R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math 132 (1998), 491-562 Zbl0919.11036MR1625724
  3. R. E. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J 97 (1999), 219-233 Zbl0967.11022MR1682249
  4. J. H. Bruinier, Borcherds products on O ( 2 , l ) and Chern classes of Heegner divisors, (January 2000) Zbl1004.11021
  5. J. H. Bruinier, Borcherds products and Chern classes of Hirzebruch-Zagier divisors, Invent. Math 138 (1999), 51-83 Zbl1011.11027MR1714336
  6. H. Cartan, Fonctions automorphes, Séminaire, Paris No. 10 (1957/58) Zbl0112.31303
  7. M. Eichler, D. Zagier, The Theory of Jacobi Forms, 55 (1985), Birkhäuser Zbl0554.10018MR781735
  8. E. Freitag, Siegelsche Modulfunktionen, (1983), Springer-Verlag Zbl0498.10016MR871067
  9. E. Freitag, Hilbert Modular Forms, (1990), Springer-Verlag Zbl0702.11029MR1050763
  10. H. Grauert, R. Remmert, Plurisubharmonische Funktionen in komplexen Räumen, Math. Zeitschr 65 (1956), 175-194 Zbl0070.30403MR81960
  11. A. Nobs, Die irreduziblen Darstellungen der Gruppen S L 2 ( ℤ p ) , insbesondere S L 2 ( ℤ 2 ) . I. Teil, Comment Math. Helvetici 51 (1976), 465-489 Zbl0346.20022MR444787
  12. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, (1971), Princeton University Press Zbl0221.10029MR314766
  13. J.-L. Waldspurger, Engendrement par des séries thêta de certains espaces de formes modulaires, Invent. Math. 50 (1979), 135-168 Zbl0393.10025MR517775
  14. G. L. Watson, Integral quadratic forms, (1960), Cambridge University Press Zbl0090.03103MR118704

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