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} and the middle cohomology of symplectic local systems on {mathcal {M}}_{g},. In genus 3, we make this explicit in a large number of cases.
Highlights
We describe for arbitrary g the precise relation between certain spaces of Teichmüller cusp forms on Mg and the middle cohomology of the standard symplectic local systems on Mg (Theorem 13.1)
The fact that in general concomitants define meromorphic modular forms that become holomorphic after multiplication with a suitable power of χ9 forces us to analyze the order of vanishing of the modular form associated to a concomitant along the hyperelliptic locus. We express this order of vanishing in terms of the order of vanishing of the concomitant along the locus of double conics in the space of ternary quartics
In Proposition 7.3 below, we show that a Teichmüller modular form extends to M3
Summary
We show how the representation theory associated to ternary quartics can be used to describe and construct all vector-valued Siegel and Teichmüller modular forms of degree 3 (Theorem 11.6). This uses the classical notion of concomitants, of which invariants, covariants, and contravariants are special cases. We describe for arbitrary g the precise relation between certain spaces of Teichmüller cusp forms on Mg and the middle cohomology of the standard symplectic local systems on Mg (Theorem 13.1).
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