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)



Nonvanishing of Fourier coefficients of modular forms

Author: Emre Alkan
Journal: Proc. Amer. Math. Soc. 131 (2003), 1673-1680
MSC (2000): Primary 11F30
Published electronically: November 6, 2002
MathSciNet review: 1953571
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $f=\sum_{n=1}^\infty a_f(n)q^n$ be a cusp form with integer weight $k\ge 2$that is not a linear combination of forms with complex multiplication. For $n\ge 1$, let

\begin{displaymath}i_f(n):=\max\{i:a_f(n+j)=0\quad\text{ for all }0\le j\le i\}. \end{displaymath}

Improving on work of Balog, Ono, and Serre we show that $i_f(n)\ll _{f,\phi}\phi(n)$ for almost all $n$, where $\phi(x)$ is any good function (e.g. such as $\log\log(x)$) monotonically tending to infinity with $x$. Using a result of Fouvry and Iwaniec, if $f$ is a weight 2 cusp form for an elliptic curve without complex multiplication, then we show for all $n$ that $i_f(n)\ll _{f,\varepsilon} n^{\frac{69}{169}+\varepsilon}$. We also obtain conditional results depending on the Generalized Riemann Hypothesis and the Lang-Trotter Conjecture.

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

  • [B-O] A. Balog and K. Ono, The Chebotarev density theorem in short intervals and some questions of Serre, J. Number Theory 91 (2001), 356-371.
  • [F-I] E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), 311-33. MR 91b:11097
  • [E] N. Elkies, Distribution of supersingular primes, Astérisque (1992), 127-132. MR 93b:11070
  • [S] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323-401. MR 83k:12011

Similar Articles

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

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

Additional Information

Emre Alkan
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Received by editor(s): January 9, 2002
Published electronically: November 6, 2002
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2002 American Mathematical Society