Nonvanishing of Fourier coefficients of modular forms

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 \[ i_f(n):=\max \{i:a_f(n+j)=0\quad \text {for all $0\le j\le i$}\}. \] 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.