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- W4313642445 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 $sigma_{1}leq mathop{mathrm{Re}} (s)leqsigma_{2}$ 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." @default.
- W4313642445 created "2023-01-07" @default.
- W4313642445 creator A5080507414 @default.
- W4313642445 date "2023-01-11" @default.
- W4313642445 modified "2023-09-30" @default.
- W4313642445 title "Rational period functions for $Gamma_{0}^{+}(2)$ with poles only in $mathbf{Q}cup {infty}$" @default.
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- W4313642445 doi "https://doi.org/10.3792/pjaa.99.002" @default.
- W4313642445 hasPublicationYear "2023" @default.
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