Abstract. We study Shimura subvarieties of Ag obtained from families of Galois coverings f : C → C ′ where C ′ is a smooth complex projective curve of genus g ′ ≥ 1 and g = g(C). We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of Ag for g ′ = 1, 2 and for all g ≥ 2, 4 and for g ′ > 2 and g ≤ 9. In [13] similar computations were done in the case g ′ = 0. Here we find 6 families of Galois coverings, all with g ′ = 1 and g = 2, 3, 4 and we show that these are the only families with g ′ = 1 satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of Ag, while the other examples arise from certain Shimura subvarieties of Ag already obtained as families of Galois coverings of P 1 in [13]. Finally we prove that if a family satisfies this sufficient condition with g ′ ≥ 1, then g ≤ 6g ′ + 1.

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