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

• Fabien Cléry ^[1] ; Carel Faber ^[2] ; Gerard van der Geer ^[3]
  1. [1] Loughborough University

     Loughborough University

     Charnwood District, Reino Unido

     
  2. [2] Utrecht University

     Utrecht University

     Países Bajos

     
  3. [3] University of Amsterdam

     University of Amsterdam

     Países Bajos

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 4, 2020
• Idioma: inglés
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• Resumen

  • 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 M¯¯¯¯¯¯g and the middle cohomology of symplectic local systems on Mg. In genus 3, we make this explicit in a large number of cases.