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Modular forms for the \(A_{1}\)-tower

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Abstract

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct \(\chi _{5}\), the cusp form of lowest weight for the group \({\text {Sp}}(2,\mathbb {Z})\). In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice \(A_{1}\) and Igusa’s form \(\chi _{5}\) appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates.

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Acknowledgements

The results of this paper are part of the author’s Ph.D. thesis. The author would like to thank the supervisors Valery Gritsenko and Aloys Krieg for their guidance and support.

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  1. Martin Woitalla

    Present address: Raytion GmbH Düsseldorf, Duesseldorf, Germany

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  1. RWTH University, Aachen, Germany

    Martin Woitalla

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  1. Martin Woitalla
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Correspondence to Martin Woitalla.

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Communicated by Ulf Kühn.

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Woitalla, M. Modular forms for the \(A_{1}\)-tower. Abh. Math. Semin. Univ. Hambg. 88, 297–316 (2018). https://doi.org/10.1007/s12188-018-0197-6

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