On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic
Model Theory in Geometry and Arithmetic May 12, 2014  May 16, 2014
Location: SLMath: Eisenbud Auditorium
v1353
Let $K$ be the function field of a smooth and proper curve $S$ over an algebraically closed field $k$ of characteristic $p>0$. Let
$A$ be an ordinary abelian variety over $K$. Suppose that the N\'eron model $\CA$ of $A$ over $S$ has some closed fibre $\CA_s$, which is
an abelian variety of $p$rank $0$. We show that in this situation the group $A(K^\perf)$ is finitely generated (thus generalizing a special case of the LangN\'eron theorem). Here $K^\perf=K^{p^{\infty}}$ is the maximal purely inseparable extension of $K$. This result implies in particular that the "full" MordellLang conjecture is verified in the situation described above. The proof relies on the theory of semistability (of vector bundles) in positive characteristic and on the existence of the compactification of the universal abelian scheme constructed by FaltingsChai.
When $A$ is an elliptic curve, this result was proven by D. Ghioca using a different method
v1353
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