Nonvanishing of Fourier coefficients of modular forms

Abstract

Let f = ∑ n = 1 ∞ a f ( n ) q n f=\sum _{n=1}^\infty a_f(n)q^n be a cusp form with integer weight k ≥ 2 k\ge 2 that is not a linear combination of forms with complex multiplication. For n ≥ 1 n\ge 1 , let \[ i f ( n ) := max { i : a f ( n + j ) = 0 for all 0 ≤ j ≤ i } . i_f(n):=\max \{i:a_f(n+j)=0\quad \text {for all $0\le j\le i$}\}. \] Improving on work of Balog, Ono, and Serre we show that i f ( n ) ≪ f , ϕ ϕ ( n ) i_f(n)\ll _{f,\phi }\phi (n) for almost all n n , where ϕ ( x ) \phi (x) is any good function (e.g. such as log ⁡ log ⁡ ( x ) \log \log (x) ) monotonically tending to infinity with x x . Using a result of Fouvry and Iwaniec, if f f is a weight 2 cusp form for an elliptic curve without complex multiplication, then we show for all n n that i f ( n ) ≪ f , ε n 69 169 + ε 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.