Geodesic
Spherical Image of Local Hecke Series of Genus Four
Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 676-680.

Voir la notice de l'article provenant de la source Math-Net.Ru

For the spherical image of a polynomial fraction, we obtain the explicit formula conjectured by Shimura in 1963 for generating Hecke series in the particular case of genus $4$. As in our previous work, we use formulas due to Andrianov for the Satake spherical map.
Keywords: Hecke operators, Dirichlet series, Euler factorization, spherical map. 
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K. A. Vankov. Spherical Image of Local Hecke Series of Genus Four. Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 676-680. http://geodesic.mathdoc.fr/item/MZM_2007_81_5_a3/

[1] A. A. Panchishkin, K. A. Vankov, “Explicit Shimura's conjecture for Sp$_3$ on a computer”, Math. Res. Lett., 2007 (to appear) ; arXiv: math.NT/0607158 | MR

[2] G. Shimura, “On modular correspondences for $Sp(n,\mathbb Z)$ and their congruence relations”, Proc. Nat. Acad. Sci. U.S.A., 49 (1963), 824–828 | DOI | MR | Zbl

[3] A. N. Andrianov, Quadratic Forms and Hecke Operators, Grundlehren Math. Wiss., 286, Berlin, Springer-Verlag, 1987 | MR | Zbl

[4] A. N. Andrianov, V. G. Zhuravlev, Modulyarnye formy i operatory Gekke, Nauka, M., 1990 | MR | Zbl

[5] A. N. Andrianov, “Sfericheskie funktsii dlya GL$_n$ nad lokalnymi polyami i summirovanie ryadov Gekke”, Matem. sb., 83 (125):3 (11) (1970), 429–451 | MR | Zbl