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Additive and Multiplicative Lifting Properties of the Igusa Modular Form

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Automorphic Forms

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 115))

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Abstract

The first cusp form χ 10 for the Siegel modular group of genus 2 is the Igusa modular form. It has been known by Gritsenko and Nikulin based on work of Borcherds that χ 10 is a Borcherds lift (multiplicative lift) and by Maass that it is a Saito–Kurokawa lift (additive lift). In this paper we show that these two properties characterize the Igusa modular form. By Bruinier, Siegel modular forms of genus 2 with Heegner divisor are Borcherds products. Hence every Saito–Kurokawa lift has a divisor different from a Heegner divisor except the lift is equal to the Igusa modular form. This implies that Siegel-type Eisenstein series do not have a Heegner divisor. Since in string theory Siegel modular forms, which are additive and multiplicative lifts play a prominent role, our uniqueness result may have some applications in this theory.

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References

  1. R.E. Borcherds, Automorphic forms on \(O_{s+2,2}(\mathbb{R})\) and infinite products, Invent. Math. 120, 161–213 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. R.E. Borcherds, Automorphic forms with singularities on Grassmannians. Invent. Math. 132, 491–562 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. J.H. Bruinier, in Borcherds Products on O(2,l) and Chern Classes of Heegner Divisors, Lecture Notes in Mathematics, vol. 1780 (Springer, New York, 2002)

    Google Scholar 

  4. M. Cheng, E. Verlinde, Wall crossing, discrete attractor flow, and borcherds algebra. SIGMA 4, (2008)

    Google Scholar 

  5. M. Cheng, E. Verlinde, Dying dyons don’t count. arXiv:0706.2363. JHEP 0709 (2007)

    Google Scholar 

  6. M. Cheng, A. Dabholkar, Borcherds-Kac-Moody symmetry of \(\mathcal{N} = 4\) dyons. Comm. Num. Theory Phys. 3(1), 59–110 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Cox, Primes of the Form x 2 + ny 2 (Wiley, New York, 1989)

    Google Scholar 

  8. A. Dabholkar, S. Murthy, D. Zagier, Quantum black holes, wall crossing, and mock modular forms. arXiv:1208.4074v1 [hep-th] 20 Aug (2012)

    Google Scholar 

  9. R. Dijkgraff, E. Verlinde, H. Verlinde, Counting dyons in N = 4 string theory. Nucl. Phys. B. 484, 543–561 (1997)

    Article  Google Scholar 

  10. M. Eichler, D. Zagier, in Theory of Jacobi Forms, Progress in Mathematics, vol. 55 (Birkhäuser, Boston, 1985)

    Google Scholar 

  11. G. van der Geer, On the geometry of a Siegel modular threefold. Math. Ann. 260, 317–350 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. G. van der Geer, in Hilbert Modular Surfaces (Springer, New York, 1988)

    MATH  Google Scholar 

  13. V. A. Gritsenko, V. V. Nikulin, The Igusa modular forms and “the simplest” Lorentzian Kac–Moody algebra. Sb. Math. 187, 1601–1643 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. V.A. Gritsenko, V.V. Nikulin, Siegel automorphic form corrections of some Lorentzian Kac-Moody Lie algebras. Am. J. Math. 119, 181–224 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. V.A. Gritsenko, V.V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebra, part II. Int. J. Math. 9, 201–275 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. B. Heim, On the spezialschar of maass. Int. J. Math. Math. Sci. 2010 (2010)

    Google Scholar 

  17. B. Heim, A. Murase, On the Igusa modular form of weight 10, RIMS Proceeding on Automorphic forms, trace formulas and zeta functions. RIMS Kokyuroku 1767, 179–187 (2011)

    Google Scholar 

  18. B. Heim, A. Murase, Borcherds lifts on \(Sp_{2}(\mathbb{Z})\), in Geometry and Analysis of Automorphic Forms of Several Variables, Proceedings of the international symposium in honor of Takayuki Oda on the Occasion of his 60th birthday (World Scientific, Singapore, 2012), pp. 56–76

    Google Scholar 

  19. J. Igusa, On Siegel modular forms of genus two. Am. J. Math. 84, 175–200 (1962)

    Article  MathSciNet  Google Scholar 

  20. J. Igusa, On Siegel modular forms of genus two (II). Am. J. Math. 86, 392–412 (1964)

    Article  MathSciNet  Google Scholar 

  21. H. Klingen, Introductory Lectures on Siegel Modular Forms (Cambridge University Press, Cambridge, 1990)

    Book  MATH  Google Scholar 

  22. S. Lang, Elliptic Functions (Springer, New York/Berlin, 1987)

    Book  MATH  Google Scholar 

  23. H. Maass, Über eine spezialschar von modulformen zweiten grades I. Invent. Math. 52, 95–104 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Maass, Über eine spezialschar von modulformen zweiten grades II. Invent. Math. 53, 249–253 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  25. H. Maass, Über eine Spezialschar von Modulformen zweiten grades III. Invent. Math. 53 255–265 (1979)

    Article  MathSciNet  Google Scholar 

  26. D. Shih, A. Strominger, X. Yin, Recounting dyons in N = 4 string theory. J. High Energy Phys. 10 (2006)

    Google Scholar 

  27. D. Zagier, Sur la conjecture de Saito-Kurokawa (d’aprè H. Maaß), in Séminaire Delange-Pisot-Poitou, Progress in Mathematics. vol. 12 (Birkhäuser, Boston, 1980), pp. 371–394

    Google Scholar 

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Acknowledgement

The authors thank the referee for many helpful comments.

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Authors and Affiliations

  1. German University of Technology in Oman, Way No. 36, Building No. 331, North Ghubrah, Muscat, Sultanate of Oman

    Bernhard Heim

  2. Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8555, Japan

    Atsushi Murase

Authors
  1. Bernhard Heim
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  2. Atsushi Murase
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Correspondence to Bernhard Heim .

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Editors and Affiliations

  1. Department of Mathematics and Science, German University of Technology in Oman, Muscat, Oman

    Bernhard Heim

  2. Department of Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman

    Mehiddin Al-Baali

  3. Department of Mathematics Graduate School of Science, Osaka University, Osaka, Japan

    Tomoyoshi Ibukiyama

  4. Department of Mathematics and Science, German University of Technology in Oman, Muscat, Oman

    Florian Rupp

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Heim, B., Murase, A. (2014). Additive and Multiplicative Lifting Properties of the Igusa Modular Form. In: Heim, B., Al-Baali, M., Ibukiyama, T., Rupp, F. (eds) Automorphic Forms. Springer Proceedings in Mathematics & Statistics, vol 115. Springer, Cham. https://doi.org/10.1007/978-3-319-11352-4_8

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