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.
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Acknowledgements
The author would like to thank Professor S. Böcherer for valuable discussions on the proof of \(h_{15}\in M_{15}^+(\Gamma _0^{(1)}(4),\chi _{-4})\). The idea of the first version of its proof using the twisting operator is due to him. The author would like to thank the referee for the comments which helped improving the manuscript. The author is supported by JSPS KAKENHI Grant Number JP18K03229.
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Communicated by Jens Funke.
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Kikuta, T. A ring of symmetric Hermitian modular forms of degree 2 with integral Fourier coefficients. Abh. Math. Semin. Univ. Hambg. 89, 209–223 (2019). https://doi.org/10.1007/s12188-019-00205-8
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DOI: https://doi.org/10.1007/s12188-019-00205-8