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\mathrm{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.