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Orders at infinity of modular forms with Heegner divisors


Authors: Carl Erickson, Alison Miller and Aaron Pixton
Journal: Proc. Amer. Math. Soc. 135 (2007), 3115-3126
MSC (2000): Primary 11F33; Secondary 11F11, 11E45
DOI: https://doi.org/10.1090/S0002-9939-07-08846-6
Published electronically: June 21, 2007
MathSciNet review: 2322741
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Abstract: Borcherds described the exponents $ a(n)$ in product expansions $ f = q^h \prod_{n = 1}^{\infty} (1-q^n)^{a(n)}$ of meromorphic modular forms with a Heegner divisor. His description does not directly give any information about $ h$, the order of vanishing at infinity of $ f$. We give $ p$-adic formulas for $ h$ in terms of generalized traces given by sums over the zeroes and poles of $ f$. Specializing to the case of the Hilbert class polynomial $ f = \mathcal H_d(j(z))$ yields $ p$-adic formulas for class numbers that generalize past results of Bruinier, Kohnen and Ono. We also give new proofs of known results about the irreducible decomposition of the supersingular polynomial $ S_p(X)$.


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Additional Information

Carl Erickson
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: cerickson@stanford.edu

Alison Miller
Affiliation: 320 Dunster House Mail Center, Cambridge, Massachusetts 02138
Email: miller5@fas.harvard.edu

Aaron Pixton
Affiliation: 741 Echo Road, Vestal, New York 13850
Email: apixton@princeton.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08846-6
Received by editor(s): June 10, 2005
Received by editor(s) in revised form: July 26, 2006
Published electronically: June 21, 2007
Communicated by: Ken Ono
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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