A ring of symmetric Hermitian modular forms of degree $2$ with integral Fourier coefficients

Abstract

We determine the structure over Z of the ring of symmetric Hermitian modular forms with respect to Q( √ −1) of degree 2 (with a character), whose Fourier coefficients are integers. Namely, we give a set of generators consisting of 24 modular forms. As an application of our structure theorem, we give the Sturm bounds of such the modular forms of weight k with 4 | k, in the case p = 2, 3. We remark that the bounds for p ≥ 5 are already known.