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The effective Chebotarev density theorem and modular forms modulo $ \mathfrak{m}$

Author: Sam Lichtenstein
Journal: Proc. Amer. Math. Soc. 136 (2008), 3419-3428
MSC (2000): Primary 11F33
Published electronically: May 7, 2008
MathSciNet review: 2415025
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Abstract: Suppose that $ f$ (resp. $ g$) is a modular form of integral (resp. half-integral) weight with coefficients in the ring of integers $ \mathcal{O}_K$ of a number field $ K$. For any ideal $ \mathfrak{m}\subset \mathcal{O}_K$, we bound the first prime $ p$ for which $ f\mid T_p$ (resp. $ g\mid T_{p^2}$) is zero ( $ \mod\mathfrak{m}$). Applications include the solution to a question of Ono (2001) concerning partitions.

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  • [AB] S. Ahlgren and M. Boylan, Arithmetic properties of the partition function, Invent. Math. (3) 153 (2003), 487-502. MR 2000466 (2004e:11115)
  • [AO] S. Ahlgren and K. Ono, Congruences and conjectures for the partition function, Proceedings of the Conference on $ q$-series with Applications to Combinatorics, Number Theory, and Physics, Contemp. Math. 291, Amer. Math. Soc., Providence, RI, 2001, 1-10. MR 1874518 (2002j:11120)
  • [At] A. O. L. Atkin, Multiplicative congruence properties and density problems for $ p(n)$, Proc. London Math. Soc. (3) 18 (1968), 563-576. MR 0227105 (37:2690)
  • [EOY] N.D. Elkies, K. Ono, and T. Yang, Reduction of CM elliptic curves and modular function congruences, Int. Math. Res. Not. 44 (2005), 2695-2707. MR 2181309 (2006k:11076)
  • [GO] L. Guo and K. Ono, The partition function and the arithmetic of certain modular $ L$-functions, Int. Math. Res. Not. 21 (1999), 1179-1197. MR 1728677 (2000m:11102)
  • [LMO] J.C. Lagarias, H.L. Montgomery and A.M. Odlyzko, A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math. (3) 54 (1979), 271-296. MR 553223 (81b:12013)
  • [Oe] J. Oesterlé, Versions effectives du théorème de Chebotarev sous l'hypothése de Riemann généralisée, Astérisque 61 (1979), 165-167.
  • [O1] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $ q$-series, CBMS 102, American Math. Soc., 2004. MR 2020489 (2005c:11053)
  • [O2] K. Ono, Distribution of the partition function modulo $ m$, Ann. of Math. 151 (2000), 293-307. MR 1745012 (2000k:11115)
  • [OSk] K. Ono and C. Skinner, Fourier coefficients of half-integral weight modular forms modulo $ \ell$, Ann. Math. (2) 147 (1998), 453-470. MR 1626761 (99f:11059a)
  • [S1] J-P. Serre, Divisibilité de certaines fonctions arithmétiques, Enseignement Math. 22 (1976), 227-260. MR 0434996 (55:7958)
  • [S2] J-P. Serre, Quelque applications du théorème de densité de Chebotarev, Publ. Math. IHES no.  54 (1981), 323-401. MR 644559 (83k:12011)
  • [St] J. Sturm, On the congruence of modular forms, Lect. Notes in Math. 1240, Springer, Berlin, 1987, 275-280. MR 894516 (88h:11031)
  • [SwD] H.P.F. Swinnerton-Dyer, On $ \ell$-adic representations and congruences for coefficients of modular forms, Lect. Notes in Math. 350, Springer, Berlin, 1973, 1-55. MR 0406931 (53:10717a)

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Additional Information

Sam Lichtenstein
Affiliation: 286 Adams House Mail Center, Harvard University, Cambridge, Massachusetts 02138

Received by editor(s): July 18, 2007
Received by editor(s) in revised form: August 25, 2007
Published electronically: May 7, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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