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)

 
 

 

$ 2$-adic properties of modular functions associated to Fermat curves


Author: Matija Kazalicki
Journal: Proc. Amer. Math. Soc. 139 (2011), 4265-4271
MSC (2000): Primary 11F03, 11F30, 11F33
DOI: https://doi.org/10.1090/S0002-9939-2011-10854-2
Published electronically: April 29, 2011
MathSciNet review: 2823072
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For an odd integer $ N$, we study the action of Atkin's $ U(2)$-operator on the modular function $ x(\tau)$ associated to the Fermat curve: $ X^N+Y^N=1$. The function $ x(\tau)$ is modular for the Fermat group $ \Phi(N)$, generically a noncongruence subgroup. If $ x(\tau)=q^{-1}+\sum_{i=1}^\infty a(iN-1)q^{iN-1}$, we essentially prove that $ \lim_{n \rightarrow 0}a(n)=0$ in the $ 2$-adic topology.


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

  • 1. S. Akiyama, On the $ 2^n$ divisibility of the Fourier coefficients of $ J_q$ functions and the Atkin conjecture for $ p=2$, Analytic number theory and related topics (Tokyo, 1991), World Sci. Publishing, 1993, 1-15. MR 1342302 (96d:11046)
  • 2. A. O. L. Atkin, J. N. O'Brien, Some properties of $ p(n)$ and $ c(n)$ modulo powers of 13, Trans. Amer. Math. Soc. 126 (1967), 442-459. MR 0214540 (35:5390)
  • 3. A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J. 8 (1967), 14-32. MR 0205958 (34:5783)
  • 4. A. O. L. Atkin, H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. of California, Los Angeles, 1968), Amer. Math. Soc., 1971, 1-25. MR 0337781 (49:2550)
  • 5. A. O. L. Atkin, W.-C. W. Li, L. Long, On Atkin-Swinnerton-Dyer congruence relations. II, Math. Ann. 340 (2008), no. 2, 335-358. MR 2368983 (2009a:11102)
  • 6. J. Lehner, Divisibility properties of the Fourier coefficients of the modular invariant $ j(\tau)$, Amer. J. Math. 71 (1949), 136-148. MR 0027801 (10:357a)
  • 7. W.-C. W. Li, L. Long, Z. Yang, Modular forms for noncongruence subgroups, Quart. J. Pure Appl. Math. 1 (2005), 205-221. MR 2155139 (2006k:11077)
  • 8. W.-C. W. Li, L. Long, Z. Yang, On Atkin-Swinnerton-Dyer congruence relations, J. Number Theory 113 (2005), no. 1, 117-148. MR 2141761 (2006c:11053)
  • 9. L. Long, On Atkin-Swinnerton-Dyer congruence relations. III, J. Number Theory 128 (2008), no. 8, 2413-2429. MR 2394828 (2009e:11085)
  • 10. D. Rohrlich, Points at infinity on the Fermat curves, Invent. Math. 39 (1977), 95-127. MR 0441978 (56:367)
  • 11. A. J. Scholl, Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences, Invent. Math. 79 (1985), 49-77. MR 774529 (86j:11045)
  • 12. A. J. Scholl, Modular forms on noncongruence subgroups, Séminaire de Théorie des Nombres, Paris 1985-86, Progr. Math., Vol. 71, Birkhäuser, Boston, MA, 1987, 199-206. MR 1017913 (90k:11049)
  • 13. A. J. Scholl, The $ l$-adic representations attached to a certain noncongruence subgroup, J. Reine Angew. Math. 392 (1988), 1-15. MR 965053 (90e:11064)
  • 14. J. G. Thompson, Hecke operators and noncongruence subgroups, Group Theory, Singapore, 1987, de Gruyter, Berlin, 1989, including a letter from J.-P. Serre, 215-224. MR 981844 (90a:20105)
  • 15. T. Yang, Cusp form of weight $ 1$ associated to Fermat curves, Duke Math. J. 83 (1996), 141-156. MR 1388846 (97e:11053)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F03, 11F30, 11F33

Retrieve articles in all journals with MSC (2000): 11F03, 11F30, 11F33


Additional Information

Matija Kazalicki
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, University of Zagreb, Bijenicka cesta 30, 10000 Zagreb, Croatia
Email: kazalick@math.wisc.edu, mkazal@math.hr

DOI: https://doi.org/10.1090/S0002-9939-2011-10854-2
Keywords: Modular forms, noncongruence subgroups, Fermat curves
Received by editor(s): April 13, 2010
Received by editor(s) in revised form: September 20, 2010, and October 23, 2010
Published electronically: April 29, 2011
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society