On the local Hecke series of some classical groups over p-adic fields

Tatsuo HINA; Takashi SUGANO

doi:10.2969/jmsj/03510133

Tatsuo HINA, Takashi SUGANO

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JOURNAL FREE ACCESS

1983 Volume 35 Issue 1 Pages 133-152

DOI https://doi.org/10.2969/jmsj/03510133

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Published: 1983 Received: September 25, 1981 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: AUTHOR Details: Wrong : Tatsuo NINA¹⁾, Takashi SUGANO¹⁾
Right : Tatsuo HINA¹⁾, Takashi SUGANO¹⁾

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) A.N. Andrianov, Shimura's conjecture for Siegel's modular group of genus 3, Soviet Math. Dokl., 8 (1967), 1474-1478.
2) A.N. Andrianov, Rationality theorems for Hecke series and zeta functions of the groups GL_n and Sp_n over local fields, Math. USSR Izv., 3 (1969), 439-476.
3) A.N. Andrianov, Spherical functions for GL_n over local fields and summation of Hecke series, Math. USSR Sb., 12 (1970), 429-452.
4) A.N. Andrianov, On factorization of Hecke polynomials for the symplectic group of genus n, Math. USSR Sb., 33 (1977), 343-373.
5) N. Bourbaki, Éléments de mathématique, Livre II Algèbre, Chap. 9, Formes sesquilinéaires et formes quadratiques, Hermann, Paris, 1959.
6) F. Bruhat, Sur les représentations des groupes classiques p-adiques I, II, Amer. J. Math., 83 (1961), 321-338, 343-368.
7) M. Eichler, Quadratische Formen und orthogonale Gruppen, Springer, Berlin-Göttingen-Heidelberg, 1952.
8) E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I, II, Math. Ann., 114 (1937), 1-28, 316-351, (Math. Werke, 789-918).
9) H. Hijikata, Maximal compact subgroups of some p-adic classical groups, 1964.
10) Y. Ihara, On certain arithmetical Dirichlet series, J. Math. Soc. Japan, 16 (1964), 214-225.
11) I. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Publ. Math. I.H.E.S., 18 (1963), 5-69.
12) G. Shimura, On modular correspondences for Sp(n, Z) and their congruence relations, Proc. Nat. Acad. Sci. U.S.A., 49 (1963), 824-828.
13) G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, 15 (1963), 33-65.
14) G. Shimura, Arithmetic of unitary groups, Ann. of Math., 79 (1964), 369-409.
15) T. Tamagawa, On the ζ-functions of a division algebra, Ann. of Math., 77 (1963), 387-405.
16) T. Tsukamoto, On the local theory of quaternionic anti-hermitian forms, J. Math. Soc. Japan, 13 (1961), 387-400.

Right : [1] A. N. Andrianov, Shimura's conjecture for Siegel's modular group of genus 3, Soviet Math. Dokl., 8 (1967), 1474-1478.
[2] A. N. Andrianov, Rationality theorems for Hecke series and zeta functions of the groups GL_n and Sp_n over local fields, Math. USSR Izv., 3 (1969), 439-476.
[3] A. N. Andrianov, Spherical functions for GL_n over local fields and summation of Hecke series, Math. USSR Sb., 12 (1970), 429-452.
[4] A. N. Andrianov, On factorization of Hecke polynomials for the symplectic group of genus n, Math. USSR Sb., 33 (1977), 343-373.
[5] N. Bourbaki, Éléments de mathématique, Livre II Algèbre, Chap. 9, Formes sesquilinéaires et formes quadratiques, Hermann, Paris, 1959.
[6] F. Bruhat, Sur les représentations des groupes classiques p-adiques I, II, Amer. J. Math., 83 (1961), 321-338, 343-368.
[7] M. Eichler, Quadratische Formen und orthogonale Gruppen, Springer, Berlin-Göttingen-Heidelberg, 1952.
[8] E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I, II, Math. Ann., 114 (1937), 1-28, 316-351, (Math. Werke, 789-918).
[9] H. Hijikata, Maximal compact subgroups of some p-adic classical groups, 1964.
[10] Y. Ihara, On certain arithmetical Dirichlet series, J. Math. Soc. Japan, 16 (1964), 214-225.
[11] I. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Publ. Math. I.H.E.S., 18 (1963), 5-69.
[12] G. Shimura, On modular correspondences for Sp(n, Z) and their congruence relations, Proc. Nat. Acad. Sci. U.S.A., 49 (1963), 824-828.
[13] G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, 15 (1963), 33-65.
[14] G. Shimura, Arithmetic of unitary groups, Ann. of Math., 79 (1964), 369-409.
[15] T. Tamagawa, On the ζ-functions of a division algebra, Ann. of Math., 77 (1963), 387-405.
[16] T. Tsukamoto, On the local theory of quaternionic anti-hermitian forms, J. Math. Soc. Japan, 13 (1961), 387-400.

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