Advances in Theoretical and Mathematical Physics

Volume 13 (2009)

Number 6

Siegel modular forms and finite symplectic groups

Pages: 1771 – 1814

DOI: http://dx.doi.org/10.4310/ATMP.2009.v13.n6.a4

Authors

Francesco Dalla Piazza

Bert van Geemen

Abstract

The finite symplectic group $Sp(2g)$ over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this representation, for various (small) values of the genus and the level, into irreducible representations. As a consequence we obtain uniqueness results for certain modular forms related to the superstring measure, a better understanding of certain modular forms in genus three studied by D'Hoker and Phong as well as a new construction of Miyawaki's cusp form of weight twelve in genus three.

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