On the non ordinary locus in Hilbert-Blumenthal surfaces
The paper studies a certain divisor, \(V\), of a moduli space for abelian surfaces with real multiplication by a real quadratic field \(L\). To be precise: It studies the complement of the ordinary locus, \(V\), in Hilbert-Blumenthal surfaces, in positive characteristic \(p\). If \(p\) is not split, \(V\) is equal to the supersingular locus. If \(p\) is not split, an explicit parameterization of \(V\) is given, yielding that it consists of rational curves, whose total number is essentially \(C\zeta_L(-1)\) \((C=1\) for \(p\) inert and \(1/2\) for \(p\) ramified). If \(p\) is inert, the singularities of \(V\) are ordinary with two branches. If \(p\) is ramified it follows from the work by \textit{P. Deligne} and \textit{G. Pappas} [Compos. Math. 90, No. 1, 59-79 (1994; Zbl 0826.14027)] that the modular surfaces is singular. The authors determine in the case at hand the exact parameterization and nature of the singularities \((p+1\) branches). For \(p\) split, it is shown that the non-ordinary locus consists of curves intersecting transversely, and that the Hecke orbit of every nonsingular point in it is dense.
moduli space for abelian surfaces with real multiplication
scientific article; zbMATH DE number 1295412