On Shimura curves in the Torelli locus of curves

Abstract

Oort has conjectured that there do not exist Shimura curves lying generically in the Torelli locus of curves of genus $g\ge 8$. We show that there do not exist one-dimensional Shimura families of semi-stable curves of genus $g\ge 5$ of Mumford type. We also show that there do not exist Shimura curves lying generically in the Torelli locus of hyperelliptic curves of genus $g\ge 8$. 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 $4$.

42 pages. Some typos are corrected. comments are welcome

References in corpus (1)

• A new inequality on the Hodge number $h^{1,1}$ of algebraic surfaces

Cited by in corpus (3)

• Curvatures of moduli space of curves and applications
• On Shimura subvarieties generated by families of abelian covers of ${\mathbb{P}}^1$
• On the slope of hyperelliptic fibrations with positive relative irregularity