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Register a new userSiegel modular forms of degree two and three and invariant theory
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Authors
Gerard van der Geer
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This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with Fabien Clery and Carel Faber.
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We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of degree three, such as dimension formulas and traces of Hecke operators. We also use it to predict several lifts from genus 1 to genus 3, as well as lifts from G_2 and new congruences of Harder type.
We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.
We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3.
We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmuller 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 Teichmuller 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.
We extend Igusas description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree two.
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