On the non ordinary locus in Hilbert-Blumenthal surfaces
type: Item
instance of: scholarly article
title: On the non ordinary locus in Hilbert-Blumenthal surfaces
zbMATH Open document ID: 0919.14014
published in: Mathematische Annalen
publication date: 1999-06-30T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11F41
zbMATH DE Number: 1295412
zbMATH Keywords: singular modular surface
zbMATH Keywords: divisor
zbMATH Keywords: moduli space for abelian surfaces with real multiplication
zbMATH Keywords: Hilbert-Blumenthal surfaces
zbMATH Keywords: positive characteristic
zbMATH Keywords: non-ordinary locus
author: Eitan Bachmat
author: Eyal Z. Goren
reviewed by: Eyal Z. Goren
MaRDI profile type: Publication
full work available at URL: s002080050270
OpenAlex ID: W2045021783
DOI: 10.1007/S002080050270
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description: scientific article; zbMATH DE number 1295412