On Vector-Valued Siegel Modular Forms of Degree 2 and Weight $(j,2)$

Abstract

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight ${\mathrm{Sym}}^j\otimes {\det }^2$ 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.