[Paper Review] The Oort conjecture on Shimura curves in the Torelli locus of curves

Xin Lü, Kang Zuo|arXiv (Cornell University)|May 19, 2014

Algebraic Geometry and Number Theory19 citations

TL;DR

This paper proves the Oort conjecture for specific classes of Shimura curves in the Torelli locus of genus-$g$ curves: those of Mumford type and those parameterizing $g$-fold self-products of elliptic curves, for $g > 11$. It further establishes non-existence for Shimura curves in the hyperelliptic Torelli locus when $g > 7$, leading to a finiteness result on smooth curves with completely decomposable Jacobians, addressing a question of Ekedahl and Serre.

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.

Motivation & Objective

• To prove the Oort conjecture that no positive-dimensional special subvarieties (specifically Shimura curves) are generically contained in the Torelli locus of genus-$g$ curves for sufficiently large $g$.
• To extend this result to Shimura curves parameterizing $g$-fold self-products of elliptic curves and to hyperelliptic curves.
• To establish a finiteness result for smooth genus-$g$ curves with completely decomposable Jacobians, addressing a question by Ekedahl and Serre.
• To apply logarithmic Higgs bundles and Arakelov-type inequalities to constrain the geometry of families of semi-stable curves.

Proposed method

• Utilizes logarithmic Higgs bundles on curves in the moduli space $\mathcal{A}_g$ to analyze the variation of Hodge structures.
• Applies strict Arakelov inequalities to families of semi-stable curves to derive geometric constraints on the existence of Shimura curves.
• Employs Miyaoka-Yau-type inequalities and sharp slope inequalities for families of semi-stable curves to bound invariants.
• Analyzes the flat part of $R^1\bar{f}_*\mathbb{Q}$ for hyperelliptic semi-stable families to study monodromy and Hodge-theoretic properties.
• Uses the involution $\sigma_g$ on the moduli space $\mathcal{S}_g$ to relate the hyperelliptic involution to the structure of the moduli space.
• Combines differential-geometric and arithmetic techniques, including the theory of special subvarieties in Shimura varieties and CM points.

Experimental results

Research questions

• RQ1 Do Shimura curves of Mumford type exist generically in the Torelli locus of genus-$g$ curves for $g > 11$?
• RQ2 Can Shimura curves parameterizing $g$-fold self-products of elliptic curves be contained generically in the Torelli locus for $g > 11$?
• RQ3 Are there any Shimura curves contained generically in the Torelli locus of hyperelliptic curves when $g > 7$?
• RQ4 What finiteness properties hold for smooth genus-$g$ curves with completely decomposable Jacobians?
• RQ5 How do strict Arakelov and Miyaoka-Yau inequalities constrain the geometry of families of semi-stable curves in $\mathcal{A}_g$?

Key findings

• For $g > 11$, there do not exist Shimura curves of Mumford type contained generically in the Torelli locus of genus-$g$ curves.
• For $g > 11$, there do not exist Shimura curves parameterizing $g$-fold self-products of elliptic curves contained generically in the Torelli locus.
• For $g > 7$, there do not exist Shimura curves contained generically in the Torelli locus of hyperelliptic curves.
• The results imply a finiteness result for smooth genus-$g$ curves whose Jacobians are isogenous to products of elliptic curves, resolving a special case of a question by Ekedahl and Serre.
• The strict Arakelov inequalities and Miyaoka-Yau-type inequalities are shown to be effective tools in obstructing the existence of such Shimura curves.
• The flat part of $R^1\bar{f}_*\mathbb{Q}$ for hyperelliptic semi-stable families is analyzed to support the non-existence results via monodromy and Hodge-theoretic constraints.