H.P.F. Swinnerton-Dyer determined the structure of the ring of modular forms modulo in the elliptic modular case. In this paper, the structure of the ring of Hilbert modular forms modulo is studied. In the case where the discriminant of corresponding quadratic field is 8 (or 5), the explicit structure is determined.
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Shoyu Nagaoka, On Hilbert modular forms modulo : explicit ring structure. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 357–368DOI 10.4171/RMI/460