The theory of Jacobi forms over the Cayley numbers

Eie, M.; Krieg, A.

doi:10.1090/S0002-9947-1994-1195510-8

The theory of Jacobi forms over the Cayley numbers
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by M. Eie and A. Krieg

Trans. Amer. Math. Soc. 342 (1994), 793-805

DOI: https://doi.org/10.1090/S0002-9947-1994-1195510-8

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Abstract:

As a generalization of the classical theory of Jacobi forms we discuss Jacobi forms on $\mathcal {H} \times {\mathbb {C}^8}$, which are related with integral Cayley numbers. Using the Selberg trace formula we give a simple explicit formula for the dimension of the space of Jacobi forms. The orthogonal complement of the space of cusp forms is shown to be spanned by certain types of Eisenstein series.

References

Walter L. Baily Jr., An exceptional arithmetic group and its Eisenstein series, Ann. of Math. (2) 91 (1970), 512–549. MR 269779, DOI 10.2307/1970636
H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561–578. MR 19111
H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert, Numbers, Graduate Texts in Mathematics, vol. 123, Springer-Verlag, New York, 1990. With an introduction by K. Lamotke; Translated from the second German edition by H. L. S. Orde; Translation edited and with a preface by J. H. Ewing; Readings in Mathematics. MR 1066206, DOI 10.1007/978-1-4612-1005-4
Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
Min King Eie, The Maass space for Cayley numbers, Math. Z. 207 (1991), no. 4, 645–655. MR 1119962, DOI 10.1007/BF02571413
Min King Eie and Aloys Krieg, The Maass space on the half-plane of Cayley numbers of degree two, Math. Z. 210 (1992), no. 1, 113–128. MR 1161173, DOI 10.1007/BF02571786
Eberhard Freitag, Hilbert modular forms, Springer-Verlag, Berlin, 1990. MR 1050763, DOI 10.1007/978-3-662-02638-0
V. A. Gritsenko, Fourier-Jacobi functions in $n$ variables, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), no. Anal. Teor. Chisel i Teor. Funktsiĭ. 9, 32–44, 187–188 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 243–252. MR 982481, DOI 10.1007/BF01303648
Martin L. Karel, Fourier coefficients of certain Eisenstein series, Ann. of Math. (2) 99 (1974), 176–202. MR 344195, DOI 10.2307/1971017
Aloys Krieg, Modular forms on half-spaces of quaternions, Lecture Notes in Mathematics, vol. 1143, Springer-Verlag, Berlin, 1985. MR 807947, DOI 10.1007/BFb0075946
Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR 1021004, DOI 10.1007/3-540-29593-3
I. I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Mathematics and its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Translated from the Russian. MR 0252690

On Maass spaces for

546

C. Ziegler, Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg 59 (1989), 191–224. MR 1049896, DOI 10.1007/BF02942329