626 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
A note on the Manin-Mumford conjecture
In the article [PR1] {\it On Hrushovski's proof of the Manin-Mumford
conjecture} (Proceedings of the ICM 2002), R. Pink and the author gave a short
proof of the Manin-Mumford conjecture, which was inspired by an earlier
model-theoretic proof by Hrushovski. The proof given in [PR1] uses a difficult
unpublished ramification-theoretic result of Serre. It is the purpose of this
note to show how the proof given in [PR1] can be modified so as to circumvent
the reference to Serre's result. J. Oesterl\'e and R. Pink contributed several
simplifications and shortcuts to this note.Comment: 11 page
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
On the Manin-Mumford and Mordell-Lang conjectures in positive characteristic
We prove that in positive characteristic, the Manin-Mumford conjecture
implies the Mordell-Lang conjecture, in the situation where the ambient variety
is an abelian variety defined over the function field of a smooth curve over a
finite field and the relevant group is a finitely generated group. In
particular, in the setting of the last sentence, we provide a proof of the
Mordell-Lang conjecture, which does not depend on tools coming from model
theory.Comment: arXiv admin note: substantial text overlap with arXiv:1103.262
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