Concomitants of ternary quartics and vector-valued Siegel and Teichmüller modular forms of genus three

Abstract

Abstract We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller 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 Teichmüller cusp forms on $$\overline{\mathcal {M}}_{g}$$ M ¯ g and the middle cohomology of symplectic local systems on $${\mathcal {M}}_{g}\,$$ M g . In genus 3, we make this explicit in a large number of cases.