Geodesic
On factorization of Hecke polynomials for the sympletic group of genus $n$
Sbornik. Mathematics, Tome 33 (1977) no. 3, pp. 343-373 
A. N. Andrianov. On factorization of Hecke polynomials for the sympletic group of genus $n$. Sbornik. Mathematics, Tome 33 (1977) no. 3, pp. 343-373. http://geodesic.mathdoc.fr/item/SM_1977_33_3_a3/
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     title = {On~factorization of {Hecke} polynomials for the sympletic group of genus~$n$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the problem of finding relations between the Fourier coefficients of Siegel modular forms which are eigenfunctions for the Hecke operators, and the corresponding eigenvalues. We obtain a qualitative solution of the problem (Theorem 8.1). The method is based on studying connections between representations of the Hecke rings of the symplectic group and the special linear group on spaces of Fourier coefficients of Siegel modular forms. Bibliography: 10 titles.

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[2] I. Satake, “Theory of spherical functions on reductive algebraic group over $p$-adic fields”, Publ. Math. IHES, 1963, no. 18 | MR

[3] G. Shimura, “Arithmetic of alternating forms and quaternion hermition forms”, J. Math. Soc. Japan, 15 (1963), 33–65 | MR | Zbl

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

[5] N. A. Zharkovskaya, “O svyazi sobstvennykh chisel operatorov Gekke i koeffitsientov Fure sobstvennykh funktsii dlya zigelevykh modulyarnykh form roda $n$”, Matem. sb., 96(133) (1975), 584–593 | Zbl

[6] T. Tamagawa, “On the $\zeta$-functions of a division algebra”, Ann. Math., 77 (1963), 387–405 | DOI | MR | Zbl

[7] A. N. Andrianov, “Eilerovy proizvedeniya, otvechayuschie modulyarnym formam Zigelya roda 2”, Uspekhi matem. nauk, XXIX:3(177) (1974), 43–110 | MR

[8] A. N. Andrianov, “Simmetricheskie kvadraty dzeta-funktsii zigelevykh modulyarnykh form roda 2”, Trudy matem. in-ta im. V. A. Steklova, CXLII (1976), 22–45 | MR

[9] R. A. Rankin, “Contribution to the theory of Ramanujan's function $\tau(n)$ and similar arithmetical functions. I, II”, Proc. Cambridge Phil. Soc., 35 (1939), 351–372 | DOI | MR | Zbl

[10] G. Shimura, “On the holomorphy of certain Dirichlet series”, Proc. London Math. Soc., 31 (1975), 79–98 | DOI | MR | Zbl