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Poincaré series and the divisors of modular forms

Author: D. Choi
Journal: Proc. Amer. Math. Soc. 138 (2010), 3393-3403
MSC (2010): Primary 11F12; Secondary 11F30
Published electronically: June 3, 2010
MathSciNet review: 2661540
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Abstract: Recently, Bruinier, Kohnen and Ono obtained an explicit description of the action of the theta-operator on meromorphic modular forms $ f$ on $ SL_2(\mathbb{Z})$ in terms of the values of modular functions at points in the divisor of $ f$. Using this result, they studied the exponents in the infinite product expansion of a modular form and recurrence relations for Fourier coefficients of a modular form. In this paper, we extend these results to meromorphic modular forms on $ \Gamma_0(N)$ for an arbitrary positive integer $ N>1$.

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  • 1. M. Abramowitz, I. Stegun, Pocketbook of Mathematical Functions, Verlag Harri Deutsch, Thun, 1984. MR 768931 (85j:00005b)
  • 2. S. Ahlgren, The theta-operator and the divisors of modular forms on genus zero subgroups, Math. Res. Lett. 10 (2003), no. 5-6, 787-798. MR 2024734 (2004m:11059)
  • 3. R. E. Borcherds, Automorphic forms on $ \mathcal{O}_{s+2,2}(\mathbb{R})$ and infinite products, Invent. Math. 120 (1995), no. 1, 161-213. MR 1323986 (96j:11067)
  • 4. J. H. Bruinier, Borcherds products on $ O(2, l)$ and Chern classes of Heegner divisors, Lect. Notes Math. 1780, Springer-Verlag, Berlin (2002). MR 1903920 (2003h:11052)
  • 5. J. H. Bruinier, J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45-90. MR 2097357 (2005m:11089)
  • 6. J. Bruinier, W. Kohnen, K. Ono, The arithmetic of the values of modular functions and the divisors of modular forms, Compos. Math. 140 (2004), no. 3, 552-566. MR 2041768 (2005h:11083)
  • 7. J. H. Bruinier, T. Yang, Twisted Borcherds products on Hilbert modular surfaces and their CM values, Amer. J. Math. 129 (2007), no. 3, 807-841. MR 2325105 (2008f:11057)
  • 8. D. Choi, On values of a modular form on $ \Gamma_0(N)$, Acta Arith. 121 (2006), no. 4, 299-311. MR 2224397 (2006m:11051)
  • 9. W. Eholzer, N.-P. Skoruppa, Product expansions of conformal characters, Phys. Lett. B 388 (1996), no. 1, 82-89. MR 1418608 (97k:81132)
  • 10. D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over $ Q$, Invent. Math. 89 (1987), no. 3, 561-567. MR 903384 (88i:11034)
  • 11. D. A. Hejhal, The Selberg Trace Formula for PSL(2,R), Lecture Notes in Mathematics 1001, Springer-Verlag (1983). MR 711197 (86e:11040)
  • 12. G. Macdonald, Symmetric functions and Hall polynomials, Second Edition, Oxford University Press, Oxford, 1995. MR 1354144 (96h:05207)
  • 13. D. Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J. 52 (1973), 133-145. MR 0337788 (49:2557)

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Additional Information

D. Choi
Affiliation: School of Liberal Arts and Sciences, Korea Aerospace University, 200-1, Hwajeon-dong, Goyang, Gyeonggi 412-791, Korea

Keywords: Borcherds exponents, Poincar\'{e} series, divisors of modular forms
Received by editor(s): April 2, 2009
Received by editor(s) in revised form: July 27, 2009
Published electronically: June 3, 2010
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society

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