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)

 
 

 

A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms


Author: Jayce Getz
Journal: Proc. Amer. Math. Soc. 132 (2004), 2221-2231
MSC (2000): Primary 11F11
DOI: https://doi.org/10.1090/S0002-9939-04-07478-7
Published electronically: March 4, 2004
Corrigendum: Proc. Amer. Math. Soc. 138 (2010), 1159
MathSciNet review: 2052397
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series $E_{k}$ in the standard fundamental domain for $\Gamma $ lie on $A:= \{ e^{i \theta } : \frac{\pi }{2} \leq \theta \leq \frac{2\pi }{3} \}$. In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc $A$. Using this result we prove a speculation of Ono, namely that the zeros of the unique ``gap function" in $M_{k}$, the modular form with the maximal number of consecutive zero coefficients in its $q$-expansion following the constant $1$, has zeros only on $A$. In addition, we show that the $j$-invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight $k$.


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

  • [AO] S. Ahlgren and K. Ono, Weierstrass points on $X_{0}(p)$ and supersingular $j$-invariants, Math. Ann. 325 (2003), 355-368. MR 2004b:11086
  • [A] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. MR 55:7892
  • [IR] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, no. 84, Springer-Verlag, New York, 1990. MR 92e:11001
  • [K] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics, no. 97, Springer-Verlag, New York, 1993. MR 94a:11078
  • [MOS] C. L. Mallows, A. M. Odlyzko, and N. J. A. Sloane, Upper Bounds for Modular Forms, Lattices, and Codes, Journal of Algebra 36 (1975), 68-76. MR 51:12711
  • [RSD] F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein series, Bull. London Math. Soc. 2 (1970), 169-170. MR 41:5298
  • [S] J.-P. Serre, Congruences et formes modulaires (d'après H. P. F. Swinnerton-Dyer), Séminaire Bourbaki 416 (1971-1972), 319-338. MR 57:5904a

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F11

Retrieve articles in all journals with MSC (2000): 11F11


Additional Information

Jayce Getz
Affiliation: 4404 South Avenue West, Missoula, Montana 59804
Email: getz@fas.harvard.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07478-7
Keywords: Modular forms
Received by editor(s): March 21, 2003
Published electronically: March 4, 2004
Additional Notes: The author thanks the University of Wisconsin at Madison for its support.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society