Surfaces Isogenous to a Product of Curves, Braid Groups and Mapping Class Groups

Penegini, Matteo

doi:10.1007/978-3-319-13862-6_9

Cited by 15 publications

(24 citation statements)

References 23 publications

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“…We also show that the families (37), (40) and (25) are not contained in the hyperelliptic locus (see 4.3, 4.5, 4.8), while (8), (22), (36) and (39) are hyperelliptic (see 4.6). We show that (25) and (38) have the same image in M 4 and in A 4 , see 4.7. We also note that in the case of family (25) every Jacobian is reducible and it is possible to identify explicitely a CM point, see 4.9.…”

Section: New Examplesmentioning

confidence: 61%

Shimura Varieties in the Torelli Locus via Galois Coverings

Frediani

Ghigi

Penegini

2015

Int Math Res Notices

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150

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Given a family of Galois coverings of the projective line, we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety of Ag. By a computer program we get the list of all families in genus g ≤ 9 satisfying our condition. There are no families with g = 8, 9, all of them are in genus g ≤ 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples found by Moonen-Oort. We get 7 new non-abelian examples.

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Section: New Examplesmentioning

confidence: 61%

Shimura Varieties in the Torelli Locus via Galois Coverings

Frediani

Ghigi

Penegini

2015

Int Math Res Notices

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“…To conclude we have to prove that these are all the critical points of ϕ. But this is actually true, because none of the stabilizer subgroups of the critical points of ψ that are mapped to the first three critical values includes any subgroup of G. Concluding, by the unicity argument 4.2, the special variety given by the family (5) gives the same special variety obtained as the family (34) of Galois coverings of P 1 via (Z/4Z × Z/2Z) ⋊ (Z/2Z) found in [13]. This family is not contained in the hyperelliptic locus (see the proof of theorem 5.3 of [13]).…”

Section: Example (5)mentioning

confidence: 94%

“…(4) gives the same subvariety as family (32) of Table 2 in [13]. (5) gives the same subvariety as family (34) of Table 2 in [13].…”

Section: Introductionmentioning

confidence: 98%

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

2015

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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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“…So the submanifold T(G, θ) is not well-defined, but the subvariety M(G, θ) is well-defined. For more details see [46,8,6].…”

Section: Special Subvarieties In the Unramified Prym Locusmentioning

confidence: 99%

Shimura curves in the Prym locus

Colombo

Frediani

Ghigi

et al. 2019

Commun. Contemp. Math.

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Abstract. We study Shimura curves of PEL type in Ag generically contained in the Prym locus. We study both the unramified Prym locus, obtained usingétale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is 1-dimensional and the quotient of the base curve by the group is P 1 . We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples gotten in this way up to genus 28. We obtain 44 Shimura curves generically contained in the unramified Prym locus and 9 families generically contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus.

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Surfaces Isogenous to a Product of Curves, Braid Groups and Mapping Class Groups

Cited by 15 publications

References 23 publications

Shimura Varieties in the Torelli Locus via Galois Coverings

Shimura Varieties in the Torelli Locus via Galois Coverings

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

Shimura curves in the Prym locus

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