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
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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.

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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