In this note we compute the tautological ring of Hilbert modular varieties at an unramified prime. This is the first computation of the tautological ring of a non-compactified Shimura variety beyond the case of the Siegel modular variety \(\mathcal {A}_{g}\) . While the method generalises that of van der Geer for \(\mathcal {A}_{g}\) , there is an added difficulty in that the highest degree socle has \(d>1\) generators rather than 1. To deal with this we prove that the d cycles obtained by taking closures of codimension one Ekedahl–Oort strata are linearly independent. In contrast, in the case of \(\mathcal {A}_{g}\) it suffices to prove that the class of the p-rank zero locus is non-zero. The limitations of this method for computing the tautological ring of other non-compactified Shimura varieties are demonstrated with an instructive example.
