We discuss and extend some of the results obtained in Arakelov inequalities and the uniformization of certain rigid Shimura varieties (math.AG/0503339), restricting ourselves to the two dimensional case, i.e. to surfaces Y mapping generically finite to the moduli stack of Abelian varieties. In particular we show that Y is a Hilber modular surfaces if and only if the dergee of the Hodge bundle satisfies the Arakelov equality. In the revised version, we corrected some minor mistakes, pointed out by the referee, and we tried to improve the presentation of the text.
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