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Journal Article

Concomitants of Ternary Quartics and Vector-valued Siegel and Teichmüller Modular Forms of Genus Three

MPS-Authors
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Cléry,  Fabien
Max Planck Institute for Mathematics, Max Planck Society;

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Faber,  Carel
Max Planck Institute for Mathematics, Max Planck Society;

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Geer,  Gerard van der
Max Planck Institute for Mathematics, Max Planck Society;

External Resource

https://doi.org/10.1007/s00029-020-00581-7
(Publisher version)

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1908.04248.pdf
(Preprint), 406KB

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Citation

Cléry, F., Faber, C., & Geer, G. v. d. (2020). Concomitants of Ternary Quartics and Vector-valued Siegel and Teichmüller Modular Forms of Genus Three. Selecta Mathematica. New Series, 26(4): 55. doi:10.1007/s00029-020-00581-7.


Cite as: https://hdl.handle.net/21.11116/0000-0007-4F62-3
Abstract
We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichm\"uller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding
modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichm\"uller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.