A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms
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- by Jayce Getz PDF
- Proc. Amer. Math. Soc. 132 (2004), 2221-2231 Request permission
Corrigendum: Proc. Amer. Math. Soc. 138 (2010), 1159-1159.
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
- Scott Ahlgren and Ken Ono, Weierstrass points on $X_0(p)$ and supersingular $j$-invariants, Math. Ann. 325 (2003), no. 2, 355–368. MR 1962053, DOI 10.1007/s00208-002-0390-9
- Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
- Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
- Neal Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. MR 1216136, DOI 10.1007/978-1-4612-0909-6
- C. L. Mallows, A. M. Odlyzko, and N. J. A. Sloane, Upper bounds for modular forms, lattices, and codes, J. Algebra 36 (1975), no. 1, 68–76. MR 376536, DOI 10.1016/0021-8693(75)90155-6
- 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 260674, DOI 10.1112/blms/2.2.169
- Jean-Pierre Serre, Congruences et formes modulaires [d’après H. P. F. Swinnerton-Dyer], Séminaire Bourbaki, 24ème année (1971/1972), Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. Exp. No. 416, pp. 319–338 (French). MR 0466020
Additional Information
- Jayce Getz
- Affiliation: 4404 South Avenue West, Missoula, Montana 59804
- Email: getz@fas.harvard.edu
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2221-2231
- MSC (2000): Primary 11F11
- DOI: https://doi.org/10.1090/S0002-9939-04-07478-7
- MathSciNet review: 2052397