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

Abstract

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.