Faber polynomials and Poincaré series (1010.2176v2)

Abstract: In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished set of generators $f_{m,2-k}$. Such weakly holomorphic modular forms have been classified in terms of finitely many Eisenstein series, the unique weight 12 newform $\Delta$, and certain Faber polynomials in the modular invariant $j(z)$, the Hauptmodul for $\SL_2(\Z)$. We employ the theory of harmonic weak Maass forms and (non-holomorphic) Maass-Poincar\'e series in order to obtain the asymptotic growth of the coefficients of these Faber polynomials. Along the way, we obtain an asymptotic formula for the partial derivatives of the Maass-Poincar\'e series with respect to $y$ as well as extending an asymptotic for the growth of the $\ell$-th repeated integral of the Gauss error function at $x$ to include $\ell\in \R$ and a wider range of $x$.