Some evidence for the Coleman–Oort conjecture

The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of A g 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...

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Bibliographic Details
Published inRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 116; no. 1
Main Authors Conti, Diego, Ghigi, Alessandro, Pignatelli, Roberto
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 2022
Springer Nature B.V
Subjects
Applications of Mathematics
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Original Paper
Theoretical
Primary 14G35
14Q05
Computational group theory
Shimura varieties
14J10
Group actions on algebraic curves
Secondary 20F99
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Summary:The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of A g 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.
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-021-01195-0