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Hi,

I am looking for reference to the fact that the moduli space of abelian varieties is not smooth in positive characteristic

Thx

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5 
 What do you mean by "the moduli space of abelian varieties" ? You need to add some extra structure for it to be representable as a scheme. If you add level $n$ structure ($n>2$) and polarisation of degree $d$, the corresponding moduli space is smooth outside the primes dividing $d$ or $n$. See for instance Chai's survey, Th. 1.4 in "Arithmetic Geometry" ed. by Cornell-Silverman. – Damian Rössler Nov 10 2011 at 10:02

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