The dimension of the space of cusp forms on the Siegel upper half plane of degree two related to a quaternion unitary group

Tsuneo ARAKAWA

doi:10.2969/jmsj/03310125

Tsuneo ARAKAWA

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JOURNAL FREE ACCESS

1981 Volume 33 Issue 1 Pages 125-145

DOI https://doi.org/10.2969/jmsj/03310125

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Published: 1981 Received: March 28, 1979 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: CITATION Details: Wrong : 1) H. Braun, Konvergenz verallgemeinerter Eisensteinscher Reihen, Math. Z., 44 (1939), 387-397.
2) U. Christian, Berechnung des Ranges der Schar der Spitzenformen zur Modulgruppe zweiten Grades and Stufe q>2, J. Reine Angew. Math., 277 (1975), 130-154.
3) U. Christian. Zur Berechnung des Ranges der Schar der Spitzenformen zur Modulgruppe zweiten Grades und Stufe q>2, J. Reine Angew. Math., 296 (1977), 108-118.
4) M. Eichler, Zur Zahlentheorie der Quaternionen Algebren, J. Reine Angew. Math., 195 (1955), 127-151.
5) H. Maass, Siegel's Modular Forms and Dirichlet Series, Lecture Notes in Math., No. 216, Springer, 1971.
6) Y. Morita, An explicit formula for the dimension of spaces of Siegel modular forms of degree two, J. Fac. Sci. Univ. Tokyo, 21 (1974), 167-248.
7) A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (1956), 47-87.
8) Séminaire H. Cartan, Fonctions automorphes, 1957-1958.
9) H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math., 77 (1963), 33-71.
10) G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, 15 (1963), 33-65.
11) T. Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo, 22 (1975), 25-65.
12) C.L. Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math., 36 (1935), 527-606.
13) C.L. Siegel, Über die Zetafunktionen indefiniter quadratischer Formen, Math. Z., 43 (1938), 682-708.
14) C.L. Siegel, Über die Zetafunktionen indefiniter quadratischer Formen II, Math. Z., 44 (1939), 398-426.
15) C.L. Siegel, Einführung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann., 116 (1939), 617-657.
16) C.L. Siegel, The average measure of quadratic forms with given determinant and signature, Ann. of Math., 45 (1944), 667-685.
17) A. Weil, Adeles and algebraic groups, Lecture notes, Institute for Advanced Study, Princeton, 1961.
18) H. Yamaguchi, The parabolic contribution to the dimension of the space of cusp forms on Siegel space of degree two, to appear.
19) T. Yamazaki, On Siegel modular forms of degree two, Amer. J. Math., 98 (1976), 39-53.

Right : [1] H. Braun, Konvergenz verallgemeinerter Eisensteinscher Reihen, Math. Z., 44 (1939), 387-397.
[2] U. Christian, Berechnung des Ranges der Schar der Spitzenformen zur Modulgruppe zweiten Grades and Stufe q>2, J. Reine Angew. Math., 277 (1975), 130-154.
[3] U. Christian. Zur Berechnung des Ranges der Schar der Spitzenformen zur Modulgruppe zweiten Grades und Stufe q>2, J. Reine Angew. Math., 296 (1977), 108-118.
[4] M. Eichler, Zur Zahlentheorie der Quaternionen Algebren, J. Reine Angew. Math., 195 (1955), 127-151.
[5] H. Maass, Siegel's Modular Forms and Dirichlet Series, Lecture Notes in Math., No. 216, Springer, 1971.
[6] Y. Morita, An explicit formula for the dimension of spaces of Siegel modular forms of degree two, J. Fac. Sci. Univ. Tokyo, 21 (1974), 167-248.
[7] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (1956), 47-87.
[8] Séminaire H. Cartan, Fonctions automorphes, 1957-1958.
[9] H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math., 77 (1963), 33-71.
[10] G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan, 15 (1963), 33-65.
[11] T. Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo, 22 (1975), 25-65.
[12] C. L. Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math., 36 (1935), 527-606.
[13] C. L. Siegel, Über die Zetafunktionen indefiniter quadratischer Formen, Math. Z., 43 (1938), 682-708.
[14] C. L. Siegel, Über die Zetafunktionen indefiniter quadratischer Formen II, Math. Z., 44 (1939), 398-426.
[15] C. L. Siegel, Einführung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann., 116 (1939), 617-657.
[16] C. L. Siegel, The average measure of quadratic forms with given determinant and signature, Ann. of Math., 45 (1944), 667-685.
[17] A. Weil, Adeles and algebraic groups, Lecture notes, Institute for Advanced Study, Princeton, 1961.
[18] H. Yamaguchi, The parabolic contribution to the dimension of the space of cusp forms on Siegel space of degree two, to appear.
[19] T. Yamazaki, On Siegel modular forms of degree two, Amer. J. Math., 98 (1976), 39-53.

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