The Oort conjecture on Shimura curves in the Torelli locus of curves

Xin Lu; Kang Zuo

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arXiv · 1405.4751

The Oort conjecture on Shimura curves in the Torelli locus of curves

Xin Lu·Kang Zuo

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Abstract

Oort has conjectured that there do not exist Shimura curves contained generically in the Torelli locus of genus-$g$ curves when $g$ is large enough. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized $g$-dimensional abelian varieties isogenous to $g$-fold self-products of elliptic curves for $g>11$. We also prove that there do not exist Shimura curves contained generically in the Torelli locus of hyperelliptic curves of genus $g>7$. As a consequence, we obtain a finiteness result regarding smooth genus-$g$ curves with completely decomposable Jacobians, which is related to a question of Ekedahl and Serre.

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BibTeX RIS

Xin Lu, Kang Zuo. 2014-08-18. The Oort conjecture on Shimura curves in the Torelli locus of curves. https://arxiv.org/abs/1405.4751

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