Unlikely intersections in the Torelli locus and the G-functions method
Unlikely intersections in the Torelli locus and the G-functions method
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Torelli 轨迹和 G 函数方法不太可能相交
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Georgios Papas
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文献类型：
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作者：
Georgios Papas
考虑在 Torelli 轨迹 $T_g\subset \mathcal{A}_g$ 中的 $\bar{\Q}$ 上定义的平滑不可约 Hodge 泛型曲线 $S$。我们根据这些曲线与 $\mathcal{A}_g$ 的 Baily-Borel 紧化边界的交点为这些曲线建立 Zilber-Pink 类型的语句。例如，当我们的曲线与该边界的 $0$ 维层相交并且 $g$ 为奇数时，我们表明曲线中只有有限多个点对应的雅可比变体是非简单的。这些结果是通过 Andr'e 的 G 函数方法在几何 Hodge 结构的 1$ 参数变化中的特殊点的高度界限的特殊情况，我们在这里将其扩展到奇权重的这种变化的设置。
Consider a smooth irreducible Hodge generic curve $S$ defined over $\bar{\Q}$ in the Torelli locus $T_g\subset \mathcal{A}_g$. We establish Zilber-Pink-type statements for such curves depending on their intersection with the boundary of the Baily-Borel compactification of $\mathcal{A}_g$. For example, when our curve intersects the $0$-dimensional stratum of this boundary and $g$ is odd, we show that there are only finitely many points in the curve for which the corresponding Jacobian variety is non-simple. These results follow as a special case of height bounds for exceptional points in $1$-parameter variations of geometric Hodge structures via Andr\'e's G-functions method, which we extend here to the setting of such variations of odd weight.