Abstract:
It is shown that the Rankin convolution of two Siegel modular forms (of which at least one is a cusp form) extends meromorphically onto the whole complex plane. In the case of the full modular group of genus $n$, the singularities of the Rankin convolution are studied to within a finite number of points, and functional equations are obtained. By means of a Tauberian theorem, a limiting relation is obtained for the weighted sum of the squares of the Fourier coefficients of a cusp form.
Bibliography: 5 titles.
\Bibitem{Kal83}
\by V.~L.~Kalinin
\paper Analytic properties of the convolution of Siegel modular forms of genus~$n$
\jour Math. USSR-Sb.
\yr 1984
\vol 48
\issue 1
\pages 193--200
\mathnet{http://mi.mathnet.ru/eng/sm2118}
\crossref{https://doi.org/10.1070/SM1984v048n01ABEH002669}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=687612}
\zmath{https://zbmath.org/?q=an:0542.10020|0524.10023}
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https://www.mathnet.ru/eng/sm2118
https://doi.org/10.1070/SM1984v048n01ABEH002669
https://www.mathnet.ru/eng/sm/v162/i2/p200
This publication is cited in the following 9 articles:
Hidenori Katsurada, Henry H. Kim, Takuya Yamauchi, “Period of the Ikeda type lift for the exceptional group of type $E_{7,3}$”, Math. Z., 302:1 (2022), 559
Shuichi HAYASHIDA, “RANKIN-SELBERG METHOD FOR JACOBI FORMS OF INTEGRAL WEIGHT AND OF HALF-INTEGRAL WEIGHT ON SYMPLECTIC GROUPS”, Kyushu J. Math., 73:2 (2019), 391
Hidenori Katsurada, Hisa-aki Kawamura, “Ikeda's conjecture on the period of the Duke–Imamoḡlu–Ikeda lift”, Proc. London Math. Soc, 2015, pdv011
Cogdell J.W., Luo W., “The Bergman Kernel and MASS Equidistribution on the Siegel Modular Variety Sp(2N)(Z)/H-N”, Forum Math., 23:1 (2011), 141–159
F.L. Chiera, “On Petersson products of not necessarily cuspidal modular forms”, Journal of Number Theory, 122:1 (2007), 13
S. Böcherer, F.L. Chiera, “Petersson Products of Singular and Almost Singular Theta Series”, manuscripta math, 115:3 (2004), 281
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