Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Indivisibility of central values of $ L$-functions for modular forms


Author: Masataka Chida
Journal: Proc. Amer. Math. Soc. 143 (2015), 2829-2840
MSC (2010): Primary 11F67; Secondary 11F37
DOI: https://doi.org/10.1090/S0002-9939-2015-12503-8
Published electronically: February 25, 2015
MathSciNet review: 3336608
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we generalize works of Kohnen and Ono (in Invent. Math., 1999) and James and Ono (in Math. Ann., 1999) on indivisibility of (the algebraic part of) central critical values of $ L$-functions to higher weight modular forms.


References [Enhancements On Off] (What's this?)

  • [1] Avner Ash and Glenn Stevens, Modular forms in characteristic $ l$ and special values of their $ L$-functions, Duke Math. J. 53 (1986), no. 3, 849-868. MR 860675 (88h:11036), https://doi.org/10.1215/S0012-7094-86-05346-9
  • [2] Jan Hendrik Bruinier, Nonvanishing modulo $ l$ of Fourier coefficients of half-integral weight modular forms, Duke Math. J. 98 (1999), no. 3, 595-611. MR 1695803 (2000d:11061), https://doi.org/10.1215/S0012-7094-99-09819-8
  • [3] Pierre Deligne, Formes modulaires et représentations $ l$-adiques, Séminaire Bourbaki. Vol. 1968/69: Exposés 347-363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp. No. 355, 139-172 (French). MR 3077124
  • [4] Solomon Friedberg and Jeffrey Hoffstein, Nonvanishing theorems for automorphic $ L$-functions on $ {\rm GL}(2)$, Ann. of Math. (2) 142 (1995), no. 2, 385-423. MR 1343325 (96e:11072), https://doi.org/10.2307/2118638
  • [5] Kevin James and Ken Ono, Selmer groups of quadratic twists of elliptic curves, Math. Ann. 314 (1999), no. 1, 1-17. MR 1689260 (2000j:11095), https://doi.org/10.1007/s002080050283
  • [6] Winfried Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237-268. MR 783554 (86i:11018), https://doi.org/10.1007/BF01455989
  • [7] Winfried Kohnen and Ken Ono, Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication, Invent. Math. 135 (1999), no. 2, 387-398. MR 1666783 (2000c:11087), https://doi.org/10.1007/s002220050290
  • [8] B. Mazur, J. Tate, and J. Teitelbaum, On $ p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1-48. MR 830037 (87e:11076), https://doi.org/10.1007/BF01388731
  • [9] Ken Ono and Christopher Skinner, Non-vanishing of quadratic twists of modular $ L$-functions, Invent. Math. 134 (1998), no. 3, 651-660. MR 1660945 (2000a:11077), https://doi.org/10.1007/s002220050275
  • [10] 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 (99f:11059a), https://doi.org/10.2307/121015
  • [11] Kenneth A. Ribet, Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 17-51. Lecture Notes in Math., Vol. 601. MR 0453647 (56 #11907)
  • [12] Kenneth A. Ribet, On $ l$-adic representations attached to modular forms. II, Glasgow Math. J. 27 (1985), 185-194. MR 819838 (88a:11041), https://doi.org/10.1017/S0017089500006170
  • [13] Jean-Pierre Serre, Congruences et formes modulaires [d'après H. P. F. Swinnerton-Dyer], Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Springer, Berlin, 1973, pp. 319-338. Lecture Notes in Math., Vol. 317 (French). MR 0466020 (57 #5904a)
  • [14] Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440-481. MR 0332663 (48 #10989)
  • [15] 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 (88h:11031), https://doi.org/10.1007/BFb0072985
  • [16] 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, 1972) Springer, Berlin, 1973, pp. 1-55. Lecture Notes in Math., Vol. 350. MR 0406931 (53 #10717a)
  • [17] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375-484 (French). MR 646366 (83h:10061)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F67, 11F37

Retrieve articles in all journals with MSC (2010): 11F67, 11F37


Additional Information

Masataka Chida
Affiliation: Graduate School of Mathematics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto, Japan, 606-8502
Email: chida@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2015-12503-8
Received by editor(s): March 15, 2014
Published electronically: February 25, 2015
Additional Notes: The author was supported in part by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society