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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 1401524
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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.

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  • [A] E. Artin, Geometric Algebra, Interscience, New York, 1957. MR 18:553c
  • [FKo1] H. Farkas and Y. Kopeliovich, New theta constant identities, Israel J. Math. 82 (1993), 133-140. MR 94k:33044
  • [FKo2] -, New theta constant identities. II, Proc. Amer. Math. Soc. 123 (1995), 1009-1020. MR 95e:11050
  • [FKK] H. Farkas, Y. Kopeliovich, and Irwin Kra, Uniformizations of modular curves, Comm. Anal. Geom. 4 (1996), 207-259. MR 97j:11019a
  • [FK1] H. Farkas and I. Kra, Automorphic forms for congruence subgroups of $SL(2,\mathbb Z)$, Israel J. Math. 82 (1993), 87-132. MR 94e:11040
  • [FK2] -, Automorphic forms for subgroups of the modular group. II, J. Analyse Math. 70 (1996), 91-156. CMP 97:11
  • [Fr1] E. Freitag, Singular theta relations, Lecture Notes in Math., vol. 1487, Springer-Verlag, 1991. MR 94b:11038
  • [Ig] J. Igusa, Theta functions, Grundlehren der Math. Wissenschaften, Bd. 194, Springer-Verlag, 1972. MR 48:3972
  • [L] S. Lang, Elliptic Functions, Addison-Wesley, 1973. MR 53:13117
  • [Me] J. Mennicke, Zur Theorie Der Siegelschen Modulgruppe, Math. Ann. 159 (1965), 115-129. MR 31:5903
  • [M] D. Mumford, Tata lectures on theta, Vol. I, Birkhäuser, 1983. MR 85h:14026
  • [M3] -, Tata lectures on theta, Vol. III, Birkhäuser, 1991. MR 93d:14065
  • [RF] H. Rauch and H. Farkas, Theta functions with applications to Riemann surfaces, Williams and Wilkins, Baltimore, MD, 1974. MR 50:4595

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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

Received by editor(s): October 17, 1995
Received by editor(s) in revised form: April 25, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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