Kumari, Moni Non-vanishing of Hilbert Poincaré series. (English) Zbl 1422.11108 J. Math. Anal. Appl. 466, No. 2, 1476-1485 (2018). Summary: We prove some non-vanishing results of Hilbert Poincaré series. We derive these results, by showing that the Fourier coefficients of Hilbert Poincaré series satisfy some nice orthogonality relations for sufficiently large weight as well as for sufficiently large level. To prove later results, we generalize a method of E. Kowalski et al. [Mathematika 57, No. 1, 31–40 (2011; Zbl 1220.11063)]. MSC: 11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 40A05 Convergence and divergence of series and sequences Keywords:Hilbert modular forms; Poincaré series; non-vanishing Citations:Zbl 1220.11063 PDFBibTeX XMLCite \textit{M. Kumari}, J. Math. Anal. Appl. 466, No. 2, 1476--1485 (2018; Zbl 1422.11108) Full Text: DOI arXiv References: [1] Freitag, E., Hilbert modular forms, (1990), Springer-Verlag Berlin · Zbl 0702.11029 [2] Gaigalas, E., Poincaré series that do not identically vanish, Litov. Mat. Sb., 26, 3, 431-434, (1986) · Zbl 0627.10015 [3] Garrett, P., Holomorphic Hilbert modular forms, Wadsworth and Books/Cole Math. Ser., (1990) · Zbl 0685.10021 [4] Kowalski, E.; Saha, A.; Tsimerman, J., A note on Fourier coefficients of Poincaré series, Mathematika, 57, 1, 31-40, (2011) · Zbl 1220.11063 [5] Lehmer, D. H., The vanishing of Ramanujan’s function \(\tau(n)\), Duke Math. J., 14, 429-433, (1947) · Zbl 0029.34502 [6] Lehner, J., On the nonvanishing of Poincaré series, Proc. Edinb. Math. Soc. (2), 23, 2, 225-228, (1980) · Zbl 0454.10014 [7] Mozzochi, C. J., On the nonvanishing of Poincaré series, Proc. Edinb. Math. Soc. (2), 32, 131-137, (1989) · Zbl 0649.10018 [8] Rankin, R. A., The vanishing of Poincaré series, Proc. Edinb. Math. Soc. (2), 23, 2, 151-161, (1980) · Zbl 0454.10013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.