The Rankin-Selberg method on congruence subgroups
This article is based upon three earlier papers, one by \textit{D. Zagier} [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 415-437 (1981; Zbl 0505.10011)], one by \textit{S. D. Gupta} and \textit{X. She} [J. Number Theory 71, 159-165 (1998; Zbl 0917.11017)] and the last one by \textit{S. D. Gupta} [J. Number Theory 62, 115-126 (1997; Zbl 0872.11024)]. The Zagier paper extends the Rankin-Selberg method to (real analytic) modular forms on the full group \(SL(2,\mathbb{Z})\) which are not of exponential decay at \(i\infty\). He achieves this by truncating the usual fundamental domain of \(SL(2,\mathbb{Z})\). The 1997 Gupta article generalizes Zagier's method to congruence subgroups \(\Gamma\) of \(SL(2,\mathbb{Z})\) ``reduced at \(i\infty\)'', that is, such that \(\Gamma\) contains translation by one. The 1998 article (by Gupta and She) obtains the Zagier result on \(SL(2,\mathbb{Z})\), while avoiding truncation of the fundamental domain, and thereby achieving simplifications in the proof. The article under review effects a melding of the 1997 and 1998 papers, deriving the generalization to congruence subgroups without truncating the fundamental domain.
scientific article; zbMATH DE number 1455226