## Contributions from conjugacy classes of regular elliptic elements in Hermitian modular groups to the dimension formula of Hermitian modular cusp forms

HTML articles powered by AMS MathViewer

- by Min King Eie
- Trans. Amer. Math. Soc.
**294**(1986), 635-645 - DOI: https://doi.org/10.1090/S0002-9947-1986-0825727-6
- PDF | Request permission

## Abstract:

The dimension of the vector space of hermitian modular cusp forms on the hermitian upper half plane can be obtained from the Selberg trace formula; in this paper we shall compute the contributions from conjugacy classes of regular elliptic elements in hermitian modular groups by constructing an orthonomal basis in a certain Hilbert space of holomorphic functions. A generalization of the main Theorem can be applied to the dimension formula of cusp forms of $SU(p, q)$. A similar theorem was given for the case of regular elliptic elements of ${\text {Sp}}(n, {\mathbf {Z}})$ in [**5**] via a different method.

## References

- Hel Braun,
*Hermitian modular functions*, Ann. of Math. (2)**50**(1949), 827–855. MR**32699**, DOI 10.2307/1969581 - Hel Braun,
*Hermitian modular functions. III*, Ann. of Math. (2)**53**(1951), 143–160. MR**39005**, DOI 10.2307/1969345
Minking Eie, - Min King Eie,
*Dimensions of spaces of Siegel cusp forms of degree two and three*, Mem. Amer. Math. Soc.**50**(1984), no. 304, vi+184. MR**749684**, DOI 10.1090/memo/0304 - Min King Eie,
*Contributions from conjugacy classes of regular elliptic elements in $\textrm {Sp}(n,\,\textbf {Z})$ to the dimension formula*, Trans. Amer. Math. Soc.**285**(1984), no. 1, 403–410. MR**748846**, DOI 10.1090/S0002-9947-1984-0748846-X
R. Godement, - Suehiro Kato,
*A dimension formula for a certain space of automorphic forms of $\textrm {SU}(p,\,1)$*, Math. Ann.**266**(1984), no. 4, 457–477. MR**735528**, DOI 10.1007/BF01458540 - Hans Maass,
*Siegel’s modular forms and Dirichlet series*, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. MR**0344198** - Sigurđur Helgason,
*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455**
George W. Machkey, - A. Selberg,
*Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series*, J. Indian Math. Soc. (N.S.)**20**(1956), 47–87. MR**88511** - Hideo Shimizu,
*On discontinuous groups operating on the product of the upper half planes*, Ann. of Math. (2)**77**(1963), 33–71. MR**145106**, DOI 10.2307/1970201 - C. L. Siegel,
*Lectures on quadratic forms*, Tata Institute of Fundamental Research Lectures on Mathematics, No. 7, Tata Institute of Fundamental Research, Bombay, 1967. Notes by K. G. Ramanathan. MR**0271028**

*Dimension formulas for the vector spaces of Siegel’s modular cusp forms of degree two and degree three*, Thesis, University of Chicago, 1982, pp. 1-246.

*Généralités sur les formes modulaires*. I, II, Séminaire Henri Cartan, 10e années, 1957, 1958. L. K. Hua,

*On the theory of functions of several complex variables*. I, II, III, Amer. Math. Soc. Transl.

**32**(1962), 163-263. —,

*Inequalities involving determinants*, Amer. Math. Soc. Transl.

**32**(1962), 265-272.

*Unitary group representation in physics, probability and number theory*, Benjamin, New York, 1978.

## Bibliographic Information

- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**294**(1986), 635-645 - MSC: Primary 11F46; Secondary 11F55, 11F72, 32N15
- DOI: https://doi.org/10.1090/S0002-9947-1986-0825727-6
- MathSciNet review: 825727