About the conjecture of Zilber-Pink
This is a survey paper from the author's talk at the 2007 Journées Arithmétiques. Let \(A\) denote a semi-abelian variety over a field \(K\) of characteristic zero, \(X\) a closed integral subscheme of \(A\) that is not contained in any translate of a genuine subgroup scheme of \(A\). The author outlines the main steps of a strategy to prove the following conjecture: For any subgroup \(\Gamma\) of finite rank in \(A(K)\), the intersection \(X(K) \cap \left( \Gamma + A^{[\dim X+1]}\right)\) is not dense in \(X\). This is equivalent to the following conjecture, attributed to Pink and Zilber: The intersection of \(X(K) \cap \bigcup G(K)\) is not dense in \(X\), where the union runs over all subgroup schemes \(G\) of \(A\) such that \(\dim X + \dim G < \dim A\). As is explained, the most difficult step appears to lie in diophantine approximation.
scientific article; zbMATH DE number 5620657
scientific article