Families of abelian varieties with many isogenous fibres
Families of abelian varieties with many isogenous fibres
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具有许多同源纤维的阿贝尔品种家族
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发表时间：
2012
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通讯作者：
M. Orr
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作者：
M. Orr
令 Z 为复数上维度为 g 的主要极化阿贝尔簇的模空间的子簇。假设 Z 包含 Zariski 稠密点集，这些点对应于单个同源类的阿贝尔变体。 Andre 和 Pink 的猜想的推广预测 Z 是一个弱特殊的子品种。当dim Z = 1时，我们使用Pila--Zannier方法和Masser--Wustholz同源定理证明了这一点。当 Hecke 轨道由 CM 点组成时，这概括了 Edixhoven 和 Yafaev 的结果；当 Hecke 轨道由 Galois 泛点组成时，这概括了 Pink 的结果。
Let Z be a subvariety of the moduli space of principally polarised abelian varieties of dimension g over the complex numbers. Suppose that Z contains a Zariski dense set of points which correspond to abelian varieties from a single isogeny class. A generalisation of a conjecture of Andre and Pink predicts that Z is a weakly special subvariety. We prove this when dim Z = 1 using the Pila--Zannier method and the Masser--Wustholz isogeny theorem. This generalises results of Edixhoven and Yafaev when the Hecke orbit consists of CM points and of Pink when it consists of Galois generic points.