A Note on the Conjectures of Andre-Oort and Pink
by Gisbert Wüstholz  

Vol. 9 No. 4 (2014) P.735~P.779

ABSTRACT

This note consists of two parts. In the first we give a -- as we believe -- more conceptual proof of a slightly sharper effective version of a very nice result published by K$\ddot{u}$hne on the Andr$\acute{e}$-Oort conjecture for curves in $\mathbb{A}^1 \times \mathbb{A}^1$. The second part deals with an extension of the Andr$\acute{e}$-Oort conjecture by Pink where Shimura varieties are replaced by mixed Shimura varieties. We consider the particular case when the mixed Shimura variety is the product of two universal elliptic curves.

KEYWORDS
Andr$\acute{e}$ - Oort conjecture, Pink conjecture, linear forms in logarithms, mixed Shimura varieties, variation of Faltings height.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 11J86, 11J89, 11G18, 14G05, 14K20, 14K22.

MILESTONES

Received: 2014-01-30
Revised : 2014-09-02
Accepted: 2014-09-02

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