The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of generically contained in the Jacobian locus. Counterexamples are known for ≤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) ≤9. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for ≤100 there are no other families than those already known.
Conti, D., Ghigi, A., Pignatelli, R. (2022). Some evidence for the Coleman–Oort conjecture. REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS, FÍSICAS Y NATURALES. SERIE A, MATEMÁTICAS, 116(1) [10.1007/s13398-021-01195-0].
Some evidence for the Coleman–Oort conjecture
Conti, D;
2022
Abstract
The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of generically contained in the Jacobian locus. Counterexamples are known for ≤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) ≤9. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for ≤100 there are no other families than those already known.
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