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.

中文翻译：

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三元四元数和向量值的Siegel和Teichmüller模三类的伴随形式

我们展示了如何使用三元四进制表示理论来构造所有向量值的Siegel模形式和3阶Teichmüller模形式。双圆锥轨迹上伴随伴随消失的顺序与消失顺序之间的关系相应的模块形式在高椭圆位上的重要作用。我们还确定\（\ overline {\ mathcal {M}} _ {g} \）上的Teichmüller尖点形式与\（{\ mathcal {M}} _ {g} \ ，\）。在属3中，我们在很多情况下都明确了这一点。