Geodesic
Rationality theorems for Hecke series and zeta functions of the groups $GL_n$ and $Sp_n$ over local fields
Izvestiya. Mathematics, Tome 3 (1969) no. 3, pp. 439-476 
A. N. Andrianov. Rationality theorems for Hecke series and zeta functions of the groups $GL_n$ and $Sp_n$ over local fields. Izvestiya. Mathematics, Tome 3 (1969) no. 3, pp. 439-476. http://geodesic.mathdoc.fr/item/IM2_1969_3_3_a1/
@article{IM2_1969_3_3_a1,
     author = {A. N. Andrianov},
     title = {Rationality theorems for {Hecke} series and zeta functions of the groups $GL_n$ and $Sp_n$ over local fields},
     journal = {Izvestiya. Mathematics},
     pages = {439--476},
     year = {1969},
     volume = {3},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1969_3_3_a1/}
}
TY  - JOUR
AU  - A. N. Andrianov
TI  - Rationality theorems for Hecke series and zeta functions of the groups $GL_n$ and $Sp_n$ over local fields
JO  - Izvestiya. Mathematics
PY  - 1969
SP  - 439
EP  - 476
VL  - 3
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/IM2_1969_3_3_a1/
LA  - en
ID  - IM2_1969_3_3_a1
ER  - 
%0 Journal Article
%A A. N. Andrianov
%T Rationality theorems for Hecke series and zeta functions of the groups $GL_n$ and $Sp_n$ over local fields
%J Izvestiya. Mathematics
%D 1969
%P 439-476
%V 3
%N 3
%U http://geodesic.mathdoc.fr/item/IM2_1969_3_3_a1/
%G en
%F IM2_1969_3_3_a1

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

We prove the rationality of multiple Hecke series and multiple zeta functions of the multiplicative groups of simple algebras over local fields, as well as the Shimura conjecture concerning the rationality of Hecke series and zeta functions of symplectic groups over local fields.

[1] Andrianov A. N., “Gipoteza Shimury dlya zigelevoi modulyarnoi gruppy roda 3”, Dokl. AN SSSR, 177:4 (1967), 755–758 | MR | Zbl

[2] Andrianov A. N., “Ratsionalnost kratnykh ryadov Gekke polnoi lineinoi gruppy i gipoteza Shimury o ryadakh Gekke simplekticheskoi gruppy”, Dokl. AN SSSR, 183:1 (1968) | MR | Zbl

[3] Gelfand I. M., Graev M. I., Pyatetskii-Shapiro I. I., Teoriya predstavlenii i avtomorfnye funktsii, Nauka, M., 1966 | MR

[4] Bruhat F., “Distributions sur un groupe localement compact et applications a l'etude des representations des groupes $\mathfrak p$-adiques”, Bull. Soc. Math. France, 89 (1961), 43–75 | MR | Zbl

[5] Bruhat F., “Sur les representations des groupes classiques $\mathfrak p$-adiques. I”, Amer. J. Math., 83 (1961), 321–338 ; “II”, 343–368 | DOI | MR | Zbl

[6] Deuring M., Algebren, Zweite, korrigierte auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete, 41, Springer-Verlag, Berlin, 1968 | MR | Zbl

[7] Maass H., “Die Primzahlen in der Theorie der Siegelschen Modulf unktionen”, Math. Ann., 124:1 (1951), 87–122 | DOI | MR | Zbl

[8] Mautner F. I., “Spherical functions over $\mathfrak p$-adic fields”, Amer. J. Math., 80 (1958), 441–457 | DOI | MR | Zbl

[9] Satake I., “Theory of spherical functions on reductive algebraic groups over $\mathfrak p$-adic fields”, Publ. Math. JHES, 18 (1963), 229–293 | MR | Zbl

[10] Shimura G., “On modular correspondences for $Sp(n,\mathbf Z)$ and their congruence relations”, Proc. Nat. Acad. Sci. USA, 49:6 (1963), 824 | DOI | MR | Zbl

[11] Shimura G., “Arithmetic of alternating form and quaternion hermitian form”, J. Math. Soc. Japan, 15:1 (1963), 33–65 | MR | Zbl

[12] Tamagawa T., “On the $\zeta$-functions of a division algebra”, Ann. Math., 77:2 (1963), 387–405 | DOI | MR | Zbl