Orders at infinity of modular forms with Heegner divisors
HTML articles powered by AMS MathViewer
- by Carl Erickson, Alison Miller and Aaron Pixton PDF
- Proc. Amer. Math. Soc. 135 (2007), 3115-3126 Request permission
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 $S_p(X)$.References
- Richard E. Borcherds, Automorphic forms on $\textrm {O}_{s+2,2}(\mathbf R)^{+}$ and generalized Kac-Moody algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 744–752. MR 1403974
- Jan H. Bruinier, Winfried Kohnen, and Ken 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, DOI 10.1112/S0010437X03000721
- Jan H. Bruinier and Ken Ono, The arithmetic of Borcherds’ exponents, Math. Ann. 327 (2003), no. 2, 293–303. MR 2015071, DOI 10.1007/s00208-003-0452-7
- Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272 (German). MR 5125, DOI 10.1007/BF02940746
- Noam D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over $\textbf {Q}$, Invent. Math. 89 (1987), no. 3, 561–567. MR 903384, DOI 10.1007/BF01388985
- M. Kaneko and D. Zagier, Supersingular $j$-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, Computational perspectives on number theory (Chicago, IL, 1995), AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI, 1998, pp. 97–126.
- Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
- Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268 (French). MR 0404145
- H. P. F. Swinnerton-Dyer, On $l$-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 1–55. MR 0406931
- Don Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998) Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 2002, pp. 211–244. MR 1977587
Additional Information
- Carl Erickson
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 818082
- ORCID: 0000-0002-1230-7574
- 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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2322741