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On the Neron models of abelian surfaces with quaternionic multiplication

Summary: On the NŽeron models of abelian surfaces with
quaternionic multiplication
Bruce W. Jordan and David R. Morrison
In this paper we construct a certain proper regular model for abelian surfaces with
quaternionic multiplication. The construction applies when the residue characteristic
p > 3 or more generally in the tamely ramified case. Our model has the property
that the NŽeron model can be recovered as its smooth locus. We therefore obtain both
the NŽeron model and a compactification of it. We are able to give a Kodaira-like
classification of the special fibers in these proper regular models. We also list the
possible groups of connected components in the NŽeron model.
To set these results in context it is necessary to review the known results on
models of elliptic curves and the situation for abelian varieties in higher dimensions.
Originally Kodaira classified minimal models of complex analytic degenerations of
elliptic curves over the unit disk in C. Minimal here means proper and regular such
that no irreducible components of the special fiber can be contracted without losing
regularity of the total space. Minimal models for complex elliptic surfaces always exist
and are unique. Kodaira organized the possible singular fibers into a classification
scheme which has been the prevailing notation ever since.
NŽeron considered elliptic curves in the final section of his seminal paper [15] in


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics