Effective lower bounds for the norm of the Poincaré $\Theta$-operator

Abstract:

Motivated by McMullen’s proof of Kra’s conjecture that the norm of the Poincaré theta operator ${\Theta _{q,\Gamma }}$ is less than $1$ for every $q$ and $\Gamma$ of finite volume, this paper provides explicit lower bounds for this norm. These bounds are sufficient to show that $\left \| {{\Theta _{q,\Gamma }}} \right \| \to 1$ for fixed $\Gamma$ as $q \to \infty$. Here the difference from $1$ is less than $O(\frac {{{{(2\pi e)}^{q - 2}}}}{{{q^{q - 2}}}})$. For $\Gamma (N) \subseteq \Gamma \subseteq {\Gamma _0}(N)$, $\left \| {{\Theta _{q,\Gamma }}} \right \| \to 1$ for fixed $q$ as $N \to \infty$. Here the difference from $1$ is $O({N^{35 - q}})$. We prove these results by estimating $\frac {{{{\left \| {{\Theta _{q,\Gamma }}({f_p})} \right \|}_{{A_q}(\Gamma )}}}} {{{{\left \| {{f_p}} \right \|}_{{A_q}}}}}$ where the ${f_p}$ are cusp forms of weight $p \leq q - 2$. (Thus such functions may in general tend to optimize ${\Theta _{q,\Gamma }}$.) In the case of the congruence subgroups they are taken to be products of $\Delta$ and Eisenstein series. Effective formulae are presented for all implied constants.