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On Shimura curves in the Torelli locus of curves

Xin Lu, Kang Zuo

Number 3
Authors Kang Zuo
Xin Lu
Year 2013

Oort has conjectured that there do not exist Shimura curves lying generically in the Torelli locus of curves of genus g8. We show that there do not exist one-dimensional Shimura families of semi-stable curves of genus g5 of Mumford type. We also show that there do not exist Shimura curves lying generically in the Torelli locus of hyperelliptic curves of genus g8. The first result proves a slightly weaker form of the conjecture for the case of Shimura curves of Mumford type. The second result proves the conjecture for the Torelli locus of hyperelliptic curves. We also present examples of Shimura curves contained generically in the Torelli locus of curves of genus 3 and 4.

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