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For an abelian variety $A$ over a number field $F$, it is believed that $A$ has good ordinary reduction at a set of places of Dirichlet density one after passing to a suitable finite extension. It is also conjectured that the minimum extension such that the $l$-adic monodromy groups are connected is sufficient.

The well-known examples include:

  1. $End(A) = \mathbb{Z}$ with the $l$-adic monodromy groups of type $A$ or $B$ (Pink's 1998 paper).

  2. Abelian varieties of small dimension.

What are the cases for which this is known?

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