I agree with Donu. Indeed, I think even the much weaker question of whether a mod-p representation of the fundamental group of the base on Sp(2g,Z/pZ) occurs as a monodromy representation might typically have a negative answer. Given such a representation rho, you get a fibration X_rho -> P^1, whose fibers are isomorphic to the moduli space of abelian g-folds with full p-level structure; this will be general type for p large. Any abelian g-fold A/C(t) with monodromy rho corresponds to a section from P^1 back to X_rho, and I don't see why there would be such a section in general.

Oh yeah, and; the answer to your second question is yes, I think. When the monodromy is of the form

I M

0 I

with M of full rank; you can construct an abelian variety over C((t)) with totally multiplicative reduction which has any desired monodromy, as in Mumford's paper "Degenerating abelian varieties...."

If M has smaller rank maybe you can just use a product of a constant a.v. with a totally multiplicative one of dimension rank(M)?

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I agree with Donu. Indeed, I think even the much weaker question of whether a mod-p representation of the fundamental group of the base on Sp(2g,Z/pZ) occurs as a monodromy representation might typically have a negative answer. Given such a representation rho, you get a fibration X_rho -> P^1, whose fibers are isomorphic to the moduli space of abelian g-folds with full p-level structure; this will be general type for p large. Any abelian g-fold A/C(t) with monodromy rho corresponds to a section from P^1 back to X_rho, and I don't see why there would be such a section in general.