The effective Chebotarev density theorem and modular forms modulo $\mathfrak m$
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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.References
- Scott Ahlgren and Matthew Boylan, Arithmetic properties of the partition function, Invent. Math. 153 (2003), no. 3, 487–502. MR 2000466, DOI 10.1007/s00222-003-0295-6
- Scott Ahlgren and Ken Ono, Congruences and conjectures for the partition function, $q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 1–10. MR 1874518, DOI 10.1090/conm/291/04889
- A. O. L. Atkin, Multiplicative congruence properties and density problems for $p(n)$, Proc. London Math. Soc. (3) 18 (1968), 563–576. MR 227105, DOI 10.1112/plms/s3-18.3.563
- Noam Elkies, Ken Ono, and Tonghai Yang, Reduction of CM elliptic curves and modular function congruences, Int. Math. Res. Not. 44 (2005), 2695–2707. MR 2181309, DOI 10.1155/IMRN.2005.2695
- Li Guo and Ken Ono, The partition function and the arithmetic of certain modular $L$-functions, Internat. Math. Res. Notices 21 (1999), 1179–1197. MR 1728677, DOI 10.1155/S1073792899000641
- J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko, A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math. 54 (1979), no. 3, 271–296. MR 553223, DOI 10.1007/BF01390234
- J. Oesterlé, Versions effectives du théorème de Chebotarev sous l’hypothése de Riemann généralisée, Astérisque 61 (1979), 165-167.
- 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
- Ken Ono, Distribution of the partition function modulo $m$, Ann. of Math. (2) 151 (2000), no. 1, 293–307. MR 1745012, DOI 10.2307/121118
- Ken Ono and Christopher Skinner, Fourier coefficients of half-integral weight modular forms modulo $l$, Ann. of Math. (2) 147 (1998), no. 2, 453–470. MR 1626761, DOI 10.2307/121015
- Jean-Pierre Serre, Divisibilité de certaines fonctions arithmétiques, Enseign. Math. (2) 22 (1976), no. 3-4, 227–260. MR 434996
- Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
- Jacob Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275–280. MR 894516, DOI 10.1007/BFb0072985
- 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
Additional Information
- Sam Lichtenstein
- Affiliation: 286 Adams House Mail Center, Harvard University, Cambridge, Massachusetts 02138
- Email: sflicht@fas.harvard.edu
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3419-3428
- MSC (2000): Primary 11F33
- DOI: https://doi.org/10.1090/S0002-9939-08-09333-7
- MathSciNet review: 2415025