On the growth of the Taylor coefficients of automorphic forms

Abstract:

The growth of the Taylor coefficients of an automorphic form of dimension -2 with respect to a Fuchsian group $\Gamma$ is related to the area integral $\smallint {\smallint _U}|F{|^s}{(1 - |z{|^2})^t}dxdy$, and it is found that these coefficients must grow faster than a power of $n$. Moreover if $F \in H(p,\Gamma )$ then these coefficients must grow slower than a different power of $n$ and, in fact, ${a_n}/n$ is square summable if either $p = 2$ or $1 < p < \infty$ and $\Gamma$ is finitely generated of the second kind.