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

Abstract

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.

Shimura curves in the Prym locus

“…contained in j(M g ) and intersecting j(M g ). See [38] for more information, [29,17,14,32,33] for some results towards the conjecture and [16,37,22,23,27,28] for counterexamples to the conjecture in low genera.…”

“…For low genera (g ≤ 7) there do exist Shimura subvarieties of A g contained in the Torelli locus. These have all been constructed as families of Jacobians of Galois coverings of P 1 and of genus one curves ( [16], [47], [37], [38], [22], [23]) [27], [28]). All these families of curves C satisfy the sufficient condition that dim(S 2 H 0 (K C )) G = dim H 0 (2K C ) G , where G is the Galois group of the covering (see [22] Theorem 3.9).…”

Explicit formulas for infinitely many Shimura curves in genus $4$

“…We announced our results at several talks starting in February 2014, including at Oberwolfach, Paris Jussieu, and Roma Tre. At the last stage of preparing our manuscript, the preprint of Frediani, Penegini, Porru [FPP15] appeared, which studies much more generally under what conditions families of covers of elliptic curves may lead to Shimura curves. They independently discovered our examples, and much more, while we are able to compute the period matrices explicitly using the Shimura description.…”

On the Bielliptic and Bihyperelliptic Loci

“…We point out that in [16] it is proven that if g ≥ 4 the bielliptic locus does not satisfy condition (*). Then our result proves that condition (*) is necessary for bielliptic curves to yield Shimura subvarieties of A g .…”