On some questions raised by Anand Pillay and Franck Benoist

Details On some questions raised by Anand Pillay and Franck Benoist On some questions raised by Anand Pillay and Franck Benoist

2011-03-14

arxiv.org/abs/1103.2625v3

Mathematics

Algebraic Geometry

This is an expanded version of "Notes on the Mordell-Lang conjecture on positive characteristic [Version III]"

Scientific paper

We prove that indefinitely $p$-divisible points on abelian varieties defined over function fields of transcendance degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is $\mZ$ then there are no indefinitely $p$-divisible points of order a power of $p$. Finally, we prove a general result on the sparsity of points in a special fibre of an abelian variety as above, which lift to highly $p$-divisible unramified points; we show how it can be used to give a new proof of the Mordell-Lang conjecture for ordinary abelian varieties.

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