Some evidence for the Coleman–Oort conjecture View Full Text


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Article Info

DATE

2021-12-16

AUTHORS

Diego Conti, Alessandro Ghigi, Roberto Pignatelli

ABSTRACT

The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of Ag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {A}}_g$$\end{document} generically contained in the Jacobian locus. Counterexamples are known for g≤7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\le 7$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\le 9$$\end{document}. By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for g≤100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\le 100$$\end{document} there are no other families than those already known. More... »

PAGES

50

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s13398-021-01195-0

DOI

http://dx.doi.org/10.1007/s13398-021-01195-0

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1143935896


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