Sharpening 'Manin-Mumford' for certain algebraic groups in dimension 2

Some Manin-Mumford Problems

On the Manin-Mumford conjecture

We present an elementary algebraic proof of the Manin-Mumford conjecture, concerning the the intersection of a subvariety X of a semiabelian variety A with the group of torsion points of A, when all data are defined over a number field. The proof is inspired by Hrushovski’s work [2], uses ideas originating in work of the author and Martin Ziegler [5], and is closely related to recent papers of ...

The Manin Conjecture in Dimension 2

Positive Characteristic Manin-mumford Theorem

We present the details of a model theoretic proof of an analogue of the Manin-Mumford conjecture for semiabelian varieties in positive characteristic. As a by-product of the proof we reduce the general positive characteristic Mordell-Lang problem to a question about purely inseparable points on subvarieties of semiabelian varieties.

Relative Manin-Mumford for semi-abelian surfaces

We show that Ribet sections are the only obstruction to the validity of the relative Manin-Mumford conjecture for one dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber-Pink conjecture for curves in a mixed Shimura variety of dimension four, as well as the study of polynomial Pell equations with non-separable discriminants.