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Nonvanishing of Fourier coefficients of modular forms

Authors:	Emre Alkan

Institution:	Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

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 , where $\phi(x)$ is any good function (e.g. such as $\log\log(x)$ ) monotonically tending to infinity with . Using a result of Fouvry and Iwaniec, if is a weight 2 cusp form for an elliptic curve without complex multiplication, then we show for all 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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