Survey on the geometric Bogomolov conjecture

This is a survey paper of the developments on the geometric Bogomolov conjecture. We explain the recent results by the author as well as previous works concerning the conjecture. This paper also includes an introduction to the height theory over function fields and a quick review on basic notions on non-archimedean analytic geometry.Comment: 57 pages. This is an expanded lecture note of a talk at "Non-archimedean analytic Geometry: Theory and Practice" (24--28 August, 2015). It has been submitted to the conference proceedings. Appendix adde

Hilbert Irreducibility above algberaic groups

The paper offers versions of Hilbert's Irreducibility Theorem for the lifting of points in a cyclic subgroup of an algebraic group to a ramified cover. A version of Bertini Theorem in this context is also obtained.Comment: 22 page