2 research outputs found
On function field Mordell-Lang and Manin-Mumford
We present a reduction of the function field Mordell-Lang conjecture to the
function field Manin-Mumford conjecture, in all characteristics, via model
theory, but avoiding recourse to the dichotomy theorems for (generalized)
Zariski structures.
In this version 2, the quantifier elimination result in positive
characteristic is extended from simple abelian varieties to all abelian
varieties, completing the main theorem in the positive characteristic case.
In version 3, some corrections are made to the proof of quantifier
elimination in positive characteristic, and the paper is substantially
reorganized.Comment: 21 page
Semiabelian varieties over separably closed fields, maximal divisible subgroups, and exact sequences
Given a separably closed field K of positive characteristic and finite degree
of imperfection we study the # functor which takes a semiabelian variety G over
K to the maximal divisible subgroup #G of G(K). We show that the # functor need
not preserve exact sequences. The main result is an example where #G does not
have "relative Morley rank", yielding a counterexample to a claim of
Hrushovski. The methods involve studying preservation of exact sequences by the
# functor as well as issues of descent. We also develop the notion of an
iterative D-structure on a group scheme over an iterative Hasse field, as well
as giving characteristic 0 versions of our results.Comment: 55 pages In this version 3, some corrections and clarifications are
made: in section 2.3 on relative Morley rank. Also in section 5.2 where more
explanation is given of D-structures in positive characteristic. In an
appendix we give a proof of the exactness of the functor taking a semiabelian
variety to its universal vectorial extensio