Cusp forms for congruence subgroups of $Sp_n(\mathbb {Z})$ and theta functions
Author:
Yaacov Kopeliovich
Journal:
Trans. Amer. Math. Soc. 350 (1998), 3107-3118
MSC (1991):
Primary 11F32
DOI:
https://doi.org/10.1090/S0002-9947-98-01820-0
MathSciNet review:
1401524
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper we use theta functions with rational characteristic to construct cusp forms for congruence subgroups $\Gamma _g(p)$ of $Sp(g,\mathbb {Z})$.The action of the quotient group $Sp(g,\mathbb {Z}_p)$ on these forms is conjugate to the linear action of $Sp(g,\mathbb {Z}_p)$ on $(\mathbb {Z}_p)^{2g}$. We show that these forms are higher-dimensional analogues of the Fricke functions.
- E. Artin, Geometric Algebra, Interscience, New York, 1957.
- Hershel M. Farkas and Yaacov Kopeliovich, New theta constant identities, Israel J. Math. 82 (1993), no. 1-3, 133–140. MR 1239047, DOI https://doi.org/10.1007/BF02808110
- Hershel M. Farkas and Yaacov Kopeliovich, New theta constant identities. II, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1009–1020. MR 1254837, DOI https://doi.org/10.1090/S0002-9939-1995-1254837-8
- Hershel M. Farkas, Yaacov Kopeliovich, and Irwin Kra, Uniformizations of modular curves, Comm. Anal. Geom. 4 (1996), no. 1-2, 207–259. MR 1393563, DOI https://doi.org/10.4310/cag.1996.v4.n2.a2
- Hershel M. Farkas and Irwin Kra, Automorphic forms for subgroups of the modular group, Israel J. Math. 82 (1993), no. 1-3, 87–131. MR 1239046, DOI https://doi.org/10.1007/BF02808109
- ---, Automorphic forms for subgroups of the modular group. II, J. Analyse Math. 70 (1996), 91–156.
- Eberhard Freitag, Singular modular forms and theta relations, Lecture Notes in Mathematics, vol. 1487, Springer-Verlag, Berlin, 1991. MR 1165941
- Jun-ichi Igusa, Theta functions, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 194. MR 0325625
- Serge Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973. With an appendix by J. Tate. MR 0409362
- J. Mennicke, Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965), 115–129 (German). MR 181676, DOI https://doi.org/10.1007/BF01360285
- David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651
- David Mumford, Tata lectures on theta. III, Progress in Mathematics, vol. 97, Birkhäuser Boston, Inc., Boston, MA, 1991. With the collaboration of Madhav Nori and Peter Norman. MR 1116553
- Harry E. Rauch and Hershel M. Farkas, Theta functions with applications to Riemann surfaces, The Williams & Wilkins Co., Baltimore, Md., 1974. MR 0352108
Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11F32
Retrieve articles in all journals with MSC (1991): 11F32
Additional Information
Yaacov Kopeliovich
Affiliation:
Department of Mathematics, University of California, Irvine, California 92717
Address at time of publication:
Department of Mathematics, Florida State University, Tallahassee, Florida 32306
Email:
kopel@math.fsu.edu
Received by editor(s):
October 17, 1995
Received by editor(s) in revised form:
April 25, 1996
Article copyright:
© Copyright 1998
American Mathematical Society
