Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

We study Shimura subvarieties ofAgobtained from familiesof Galois coveringsf:C→C′whereC′is a smooth complex projectivecurve of genusg′≥1 andg=g(C). We give the complete list of all suchfamilies that satisfy a simple sufficient condition that ensures that theclosure of the image of the family via the Torelli map yields aShimurasubvariety ofAgforg′= 1,2 and for allg≥2,4 and forg′>2 andg≤9. In [13] similar computations were done in the caseg′= 0. Herewe find 6 families of Galois coverings, all withg′= 1 andg= 2,3,4and we show that these are the only families withg′= 1 satisfying thissufficient condition. We show that among these examples two familiesyield new Shimura subvarieties ofAg, while the other examples arisefrom certain Shimura subvarieties ofAgalready obtained as families ofGalois coverings ofP1in [13]. Finally we prove that if a family satisfiesthis sufficient condition withg′≥1, theng≤6g′+ 1.