In this paper, we will first investigate the linear relations of a one parameter family of Siegel Poincaré series. Then we give the applications to the non-vanishing of Fourier coefficients of Siegel cusp eigenforms and the central values.
Dans cet article, nous étudions d’abord les relations linéaires des familles à un paramètre des séries de Siegel–Poincaré. Nous donnons ensuite des applications de nos résultats à la non-annulation des coefficients de Fourier des formes propres cuspidales de Siegel et des valeurs centrales.
Revised:
Accepted:
Published online:
Keywords: Siegel Poincaré series, Fourier coefficients of Siegel cusp forms, non-vanishing of central values, Böcherer conjecture.
@article{JTNB_2022__34_3_755_0, author = {Zhining Wei}, title = {Linear {Relations} of {Siegel} {Poincar\'e} {Series} and {Non-vanishing} of the {Central} {Values} of {Spinor} $L$-functions}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {755--766}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {34}, number = {3}, year = {2022}, doi = {10.5802/jtnb.1226}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1226/} }
TY - JOUR AU - Zhining Wei TI - Linear Relations of Siegel Poincaré Series and Non-vanishing of the Central Values of Spinor $L$-functions JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 755 EP - 766 VL - 34 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1226/ DO - 10.5802/jtnb.1226 LA - en ID - JTNB_2022__34_3_755_0 ER -
%0 Journal Article %A Zhining Wei %T Linear Relations of Siegel Poincaré Series and Non-vanishing of the Central Values of Spinor $L$-functions %J Journal de théorie des nombres de Bordeaux %D 2022 %P 755-766 %V 34 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1226/ %R 10.5802/jtnb.1226 %G en %F JTNB_2022__34_3_755_0
Zhining Wei. Linear Relations of Siegel Poincaré Series and Non-vanishing of the Central Values of Spinor $L$-functions. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 755-766. doi : 10.5802/jtnb.1226. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1226/
[1] Euler products corresponding to Siegel modular forms of genus 2, Usp. Mat. Nauk, Volume 29 (1974) no. 3, pp. 43-110 | Zbl
[2] Quadratic forms and Hecke operators, Grundlehren der Mathematischen Wissenschaften, 286, Springer, 1987, xii+374 pages | DOI | Zbl
[3] Spectral summation formula for GSp(4) and moments of spinor -functions, J. Eur. Math. Soc., Volume 21 (2019) no. 6, pp. 1751-1774 | DOI | Zbl
[4] Bemerkungen über die Dirichletreihen von Koecher und Maass, Mathematica Gottingensis, 68, 1986, 36 pages
[5] Nonvanishing of Siegel-Poincaré series II, Acta Arith., Volume 156 (2012) no. 1, pp. 75-81 | Zbl
[6] Nonvanishing of Siegel Poincaré series, Math. Z., Volume 272 (2012) no. 3, pp. 3-4 | Zbl
[7] Explicit refinements of Böcherer’s conjecture for Siegel modular forms of squarefree level, J. Math. Soc. Japan, Volume 72 (2020) no. 1, pp. 251-301 | Zbl
[8] Refined global Gross-Prasad conjecture on special Bessel periods and Böcherer’s conjecture, J. Eur. Math. Soc., Volume 23 (2021) no. 4, pp. 1295-1331 | DOI | Zbl
[9] Table of Integrals, Series, and Products, Academic Press Inc., 2007
[10] Analytic number theory, Colloquium Publications, 53, American Mathematical Society, 2004, xii+615 pages
[11] The non-vanishing of central values of automorphic -functions and Landau-Siegel zeros, Isr. J. Math., Volume 120 (2000), pp. 155-177 | DOI
[12] Fourier coefficients of Siegel cusp forms of degree two, Nagoya Math. J., Volume 93 (1984), pp. 149-171 | DOI | Zbl
[13] Introductory lectures on Siegel modular forms, Cambridge Studies in Advanced Mathematics, 20, Cambridge University Press, 1990, x+162 pages | DOI
[14] Local spectral equidistribution for Siegel modular forms and applications, Compos. Math., Volume 148 (2012) no. 2, pp. 335-384 | DOI | Zbl
[15] A short course in automorphic functions, Holt, Rinehart and Winston, 1966
[16] On the nonvanishing of Poincaré series, Proc. Edinb. Math. Soc., II. Ser., Volume 32 (1989) no. 1, pp. 133-137
[17] Über eine Metrisierung der Automorphen Formen und die Theorie der Poincaréschen Reihen, Math. Ann., Volume 117 (1940), pp. 453-529 | DOI | Zbl
[18] Siegel modular forms. A classical and representation-theoretic approach, Lecture Notes in Mathematics, 2240, Springer, 2019, ix+138 pages | DOI
[19] Transfer of Siegel cusp forms of degree 2, Memoirs of the American Mathematical Society, 1090, American Mathematical Society, 2014, vi+107 pages
[20] The vanishing of Poincaré series, Proc. Edinb. Math. Soc., II. Ser., Volume 23 (1980), pp. 151-161 | DOI | Zbl
Cited by Sources: