Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Computing the arithmetic genus of Hilbert modular fourfolds


Authors: H. G. Grundman and L. E. Lippincott
Journal: Math. Comp. 75 (2006), 1553-1560
MSC (2000): Primary 11F41, 14E08; Secondary 14J10, 14J35
Published electronically: March 21, 2006
MathSciNet review: 2219045
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Abstract: The Hilbert modular fourfold determined by the totally real quartic number field $ k$ is a desingularization of a natural compactification of the quotient space $ \Gamma_k \backslash {\mathcal H}^4$, where $ \Gamma_k=$PSL$ _2({\mathcal O}_k)$ acts on $ {\mathcal H}^4$ by fractional linear transformations via the four embeddings of $ k$ into $ \bf R$. The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight $ (2,2,2,2)$, is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results.


References [Enhancements On Off] (What's this?)

  • 1. C. Batut, K. Belabas, D. Benardi, H. Cohen, and M. Olivier. User's Guide to PARI-GP, 1998. $ \langle$ftp://megrez.math.u-bordeaux.fr/pub/pari$ \rangle$.
  • 2. J. Buchmann, F. Diaz y Diaz, D. Ford, P. Létard, M. Olivier, M. Pohst, and A. Schwarz. Tables of number fields of low degree, $ \langle$ftp://megrez.math.u-bordeaux.fr/pub/ numberfields/$ \rangle$.
  • 3. M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR 1484479, 10.1006/jsco.1996.0126
  • 4. Eberhard Freitag, Hilbert modular forms, Springer-Verlag, Berlin, 1990. MR 1050763
  • 5. H. G. Grundman, Hilbert modular varieties of Galois quartic fields, J. Number Theory 63 (1997), no. 1, 47–58. MR 1438648, 10.1006/jnth.1997.2074
  • 6. H. G. Grundman and L. E. Lippincott, Hilbert modular fourfolds of arithmetic genus one, High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun., vol. 41, Amer. Math. Soc., Providence, RI, 2004, pp. 217–226. MR 2076248
  • 7. Friedrich E. P. Hirzebruch, Hilbert modular surfaces, Enseignement Math. (2) 19 (1973), 183–281. MR 0393045
  • 8. Friedrich Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Translated from the German and Appendix One by R. L. E. Schwarzenberger; With a preface to the third English edition by the author and Schwarzenberger; Appendix Two by A. Borel; Reprint of the 1978 edition. MR 1335917
  • 9. F. Hirzebruch and A. Van de Ven, Hilbert modular surfaces and the classification of algebraic surfaces, Invent. Math. 23 (1974), 1–29. MR 0364262
  • 10. F. Hirzebruch and D. Zagier, Classification of Hilbert modular surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 43–77. MR 0480356
  • 11. Carl Ludwig Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969), 87–102 (German). MR 0252349
  • 12. Gerard van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. MR 930101
  • 13. Don Zagier, On the values at negative integers of the zeta-function of a real quadratic field, Enseignement Math. (2) 22 (1976), no. 1-2, 55–95. MR 0406957

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Additional Information

H. G. Grundman
Affiliation: Bryn Mawr College, 101 N. Merion Ave., Bryn Mawr, Pennsylvania 19010
Email: grundman@brynmawr.edu

L. E. Lippincott
Affiliation: Bryn Mawr College, 101 N. Merion Ave., Bryn Mawr, Pennsylvania 19010
Email: llippinc@brynmawr.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01842-4
Received by editor(s): April 23, 2004
Received by editor(s) in revised form: May 10, 2005
Published electronically: March 21, 2006
Additional Notes: The first author wishes to acknowledge the support of the Faculty Research Fund of Bryn Mawr College.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.