Metric number theory of Fourier coefficients of modular forms
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Abstract:
We discuss the approximation of real numbers by Fourier coefficients of newforms, following recent work of Alkan, Ford, and Zaharescu. The main tools used here, besides the (now proved) Sato-Tate Conjecture, come from metric number theory.References
- Emre Alkan, Glyn Harman, and Alexandru Zaharescu, Diophantine approximation with mild divisibility constraints, J. Number Theory 118 (2006), no. 1, 1–14. MR 2220258, DOI 10.1016/j.jnt.2005.08.001
- Emre Alkan, Kevin Ford, and Alexandru Zaharescu, Diophantine approximation with arithmetic functions. I, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2263–2275. MR 2471917, DOI 10.1090/S0002-9947-08-04822-8
- Emre Alkan, Kevin Ford, and Alexandru Zaharescu, Diophantine approximation with arithmetic functions. II, Bull. Lond. Math. Soc. 41 (2009), no. 4, 676–682. MR 2521363, DOI 10.1112/blms/bdp051
- Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 29–98. MR 2827723, DOI 10.2977/PRIMS/31
- Alina Bucur and Kiran S. Kedlaya, An application of the effective Sato-Tate conjecture, Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemp. Math., vol. 663, Amer. Math. Soc., Providence, RI, 2016, pp. 45–56. MR 3502938, DOI 10.1090/conm/663/13349
- J. B. Conrey, W. Duke, and D. W. Farmer, The distribution of the eigenvalues of Hecke operators, Acta Arith. 78 (1997), no. 4, 405–409. MR 1438595, DOI 10.4064/aa-78-4-405-409
- S. G. Dani, On badly approximable numbers, Schmidt games and bounded orbits of flows, Number theory and dynamical systems (York, 1987) London Math. Soc. Lecture Note Ser., vol. 134, Cambridge Univ. Press, Cambridge, 1989, pp. 69–86. MR 1043706, DOI 10.1017/CBO9780511661983.006
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
- 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
- Francesc Fité, Equidistribution, $L$-functions, and Sato-Tate groups, Trends in number theory, Contemp. Math., vol. 649, Amer. Math. Soc., Providence, RI, 2015, pp. 63–88. MR 3415267, DOI 10.1090/conm/649/13020
- Etienne Fouvry and M. Ram Murty, On the distribution of supersingular primes, Canad. J. Math. 48 (1996), no. 1, 81–104. MR 1382477, DOI 10.4153/CJM-1996-004-7
- Michael Fuchs and Dong Han Kim, On Kurzweil’s 0-1 law in inhomogeneous Diophantine approximation, Acta Arith. 173 (2016), no. 1, 41–57. MR 3494133, DOI 10.4064/aa8219-1-2016
- E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 6 (1920), no. 1-2, 11–51 (German). MR 1544392, DOI 10.1007/BF01202991
- Haruzo Hida and Yoshitaka Maeda, Non-abelian base change for totally real fields, Pacific J. Math. Special Issue (1997), 189–217. Olga Taussky-Todd: in memoriam. MR 1610859, DOI 10.2140/pjm.1997.181.189
- Loo Keng Hua, Introduction to number theory, Springer-Verlag, Berlin-New York, 1982. Translated from the Chinese by Peter Shiu. MR 665428
- J. Kurzweil, On the metric theory of inhomogeneous diophantine approximations, Studia Math. 15 (1955), 84–112. MR 73654, DOI 10.4064/sm-15-1-84-112
- A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277. MR 963118, DOI 10.1007/BF02126799
- V. Kumar Murty, Explicit formulae and the Lang-Trotter conjecture, Rocky Mountain J. Math. 15 (1985), no. 2, 535–551. Number theory (Winnipeg, Man., 1983). MR 823264, DOI 10.1216/RMJ-1985-15-2-535
- M. Ram Murty, V. Kumar Murty, and N. Saradha, Modular forms and the Chebotarev density theorem, Amer. J. Math. 110 (1988), no. 2, 253–281. MR 935007, DOI 10.2307/2374502
- Jeremy Rouse and Jesse Thorner, The explicit Sato-Tate conjecture and densities pertaining to Lehmer-type questions, Trans. Amer. Math. Soc. 369 (2017), no. 5, 3575–3604. MR 3605980, DOI 10.1090/tran/6793
- Wolfgang M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178–199. MR 195595, DOI 10.1090/S0002-9947-1966-0195595-4
- Jean-Pierre Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke $T_p$, J. Amer. Math. Soc. 10 (1997), no. 1, 75–102 (French). MR 1396897, DOI 10.1090/S0894-0347-97-00220-8
- Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
- Jimmy Tseng, Badly approximable affine forms and Schmidt games, J. Number Theory 129 (2009), no. 12, 3020–3025. MR 2560849, DOI 10.1016/j.jnt.2009.05.006
- Jimmy Tseng, On circle rotations and the shrinking target properties, Discrete Contin. Dyn. Syst. 20 (2008), no. 4, 1111–1122. MR 2379490, DOI 10.3934/dcds.2008.20.1111
Additional Information
- Paloma Bengoechea
- Affiliation: Department of Mathematics, ETH Zurich, Ramistrasse 101, 8092 Zurich, Switzerland
- MR Author ID: 1085148
- Email: paloma.bengoechea@math.ethz.ch
- Received by editor(s): July 28, 2018
- Received by editor(s) in revised form: October 29, 2018
- Published electronically: March 21, 2019
- Additional Notes: The author’s research was supported by SNF grant 173976.
- Communicated by: Amanda Folsom
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2835-2845
- MSC (2010): Primary 11F30, 11K60; Secondary 11N64, 11J83
- DOI: https://doi.org/10.1090/proc/14500
- MathSciNet review: 3973887