Tautological rings of Hilbert modular varieties

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.