The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of Ag generically contained in the Jacobian locus. Counterexamples are known for g ≤ 7. They can all be constructed using families of Galois coverings of curves satisfying a numerical condition. These families are already classified in cases where: (a) the Galois group is cyclic, (b) it is abelian and the family is 1-dimensional, or c) g ≤ 9. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for g ≤ 100 there are no other families than those already known.
Some evidence for the Coleman–Oort conjecture / Conti, Diego; Ghigi, Alessandro; Pignatelli, Roberto. - In: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS, FÍSICAS Y NATURALES. SERIE A, MATEMÁTICAS. - ISSN 1578-7303. - STAMPA. - 116:4(2022), pp. 5001-5019. [10.1007/s13398-021-01195-0]
Some evidence for the Coleman–Oort conjecture
Pignatelli, Roberto
2022-01-01
Abstract
The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of Ag generically contained in the Jacobian locus. Counterexamples are known for g ≤ 7. They can all be constructed using families of Galois coverings of curves satisfying a numerical condition. These families are already classified in cases where: (a) the Galois group is cyclic, (b) it is abelian and the family is 1-dimensional, or c) g ≤ 9. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for g ≤ 100 there are no other families than those already known.File | Dimensione | Formato | |
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