Mathematics – Number Theory
Scientific paper
2004-08-30
J. Reine Angew. Math. 618 (2008), pp. 51--75
Mathematics
Number Theory
24 pages, final version accepted for publication in Crelle
Scientific paper
10.1515/CRELLE.2008.033
Let $(G,X)$ be a Shimura pair of Hodge type such that $G$ is the Mumford--Tate group of some elements of $X$. We assume that for each simple factor $G_0$ of $G^{\ad}$ there exists a simple factor of $G_{0\dbR}$ which is compact. Let $N\Ge 3$. We show that for many compact open subgroups $K$ of $G(\dbA_f)$, the Shimura variety $\Sh(G,X)/K$ has a projective integral model $\scrN$ over $\dbZ[{1\over N}]$ which is a finite scheme over a certain Mumford moduli scheme $\scrA_{g,1,N}$. Equivalently, we show that if $A$ is an abelian variety over a number field and if the Mumford--Tate group of $A_{\dbC}$ is $G$, then $A$ has potentially good reduction everywhere. The last result represents significant progress towards the proof of a conjecture of Morita. If $\scrN$ is smooth over $\dbZ[{1\over N}]$, then it is a N\'eron model of its generic fibre. In this way one gets in arbitrary mixed characteristic, the very first examples of general nature of projective N\'eron models whose generic fibres are not finite schemes over abelian varieties.
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