The Mordell-Lang problem modulo certain abelian subvarieties. (Problème de Mordell-Lang modulo certaines sous-variétés abéliennes.) (French) Zbl 1072.11038

Rémond, Gaël; Viada, Evelina

Int. Math. Res. Not. 2003, No. 35, 1915-1931 (2003).

Summary: Following a result of E. Bombieri, D. Masser and U. Zannier [Int. Math. Res. Not. 1999, 1119–1140 (1999; Zbl 0938.11031)] on tori, the second author proved that the intersection of a transversal curve \(C\) in a power \(E^g\) of a CM elliptic curve with the union of all algebraic subgroups of \(E^g\) of codimension 2 is finite. Here transversal means that \(C\) is not contained in any translate of an algebraic subgroup of codimension 1. We merge this result with G. Faltings’ [Ann. Math. (2) 133, 549–576 (1991; Zbl 0734.14007)] theorem that \(C\cap \Gamma\) is finite when \(\Gamma\) is a finite rank subgroup of \(E^g\). We obtain the finiteness of the intersection of \(C\) with the union of all \(\Gamma+B\) for \(B\) an abelian subvariety of codimension 2. As a corollary, we generalize the previous result to a curve \(C\) not contained in any proper algebraic subgroup, but possibly contained in a translate. We also have weaker analog results in the non CM case.

MSC:

11G10	Abelian varieties of dimension \(> 1\)
11G50	Heights
11J95	Results involving abelian varieties
14G25	Global ground fields in algebraic geometry
14K15	Arithmetic ground fields for abelian varieties

Citations:

Zbl 0938.11031; Zbl 0734.14007