## Siegel modular forms and finite symplectic groups.(English)Zbl 1200.81091

Summary: 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.

### MSC:

 81S10 Geometry and quantization, symplectic methods 81T70 Quantization in field theory; cohomological methods 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T60 Supersymmetric field theories in quantum mechanics

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