Concomitants of Ternary Quartics and Vector-valued Siegel and Teichm\"uller Modular Forms of Genus Three
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.
Keywords
Cite
@article{arxiv.1908.04248,
title = {Concomitants of Ternary Quartics and Vector-valued Siegel and Teichm\"uller Modular Forms of Genus Three},
author = {Fabien Cléry and Carel Faber and Gerard van der Geer},
journal= {arXiv preprint arXiv:1908.04248},
year = {2020}
}
Comments
34 pages. In an appendix (joint work of G. Farkas, R. Pandharipande, and the second author) it is shown that Teichm\"uller modular forms extend to \overline{M}_g for g at least 3. Other minor changes