Rational period functions for $\Gamma_{0}^{+}(2)$ with poles only in $\mathbf{Q}\cup \{\infty\}$

Abstract

We extend results of Knopp in [9] to the higher level case. In precise, we characterize a rational period function $q(z)$ for $\Gamma_{0}^{+}(2)$ of which poles lie only in $\mathbf{Q} \cup \{\infty\}$. We prove that the Mellin transform $\Phi_{F}(s)$ of an entire modular integral $F$ of weight $2k$ for such a rational period function $q(z)$ has an analytic continuation to the entire $s$-plane, except for possible simple poles at some rational integers, satisfies the functional equation $\Phi_{F}(2k-s) = (-1)^{k} 2^{s-k} \Phi_{F}(s)$, and is bounded on each “truncated strip” of the from $\sigma_{1}\leq \mathop{\mathrm{Re}} (s)\leq\sigma_{2}$ and $|{\mathop{\mathrm{Im}}}(s)|\geq t_{0}>0$. We also show that the converse is true. The case for $\Gamma_{0}^{+}(3)$ is addressed similarly.