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

We determine the structure over \(\mathbb {Z}\) of a ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients whose weights are multiples of 4 when the base field is the Gaussian number field \(\mathbb {Q}(\sqrt{-1})\). Namely, we give a set of generators consisting of 24 modular forms. As an application of our structure theorem, we give the Sturm bounds for such Hermitian modular forms of weight k with \(4\mid k\), for \(p=2\), 3. We remark that the bounds for \(p\ge 5\) are already known.