Thursday 27 Sep 2018NT Seminar: On a conjecture of Esnault and Langer

Damian Rossler - University of Oxford

H103 14:30-16:30


Suppose that $l_0$ is an algebraically closed field of characteristic $p>0$ and that $L$ is the function field of a variety over $l_0$. Let $A$ be


an abelian variety over $L$. There is an infinite sequence 


$$


\cdots\stackrel{V^{(p^3)}}{\to} A^{(p^2)}\stackrel{V^{(p^2)}}{\to} A^{(p)}\stackrel{V^{(p)}}{\to} A


$$


where the connecting morphisms $V^{(p^n)}}$ are the so-called Verschiebung morphisms.


Suppose now given a sequence of points $x_n\in A^{(p^n)}(L)$ and suppose 


that for all $n$ we have $V^{(p^n)}}(x_n)=x_{n-1}$. Esnault and Langer conjectured 


that $x_0$ must then be a torsion point of order prime to $p$. 


We shall prove this conjecture when $l_0=\bar\mF_p$. 


 


Our method is based on a geometric description of the Selmer group of the relative 


Frobenius morphism on abelian varieties and on the fact that finite flat group schemes of 


height one on curves are constrained by the numerical properties of the Harder-Narasimhan filtration of 


their $p$-Lie algebras.

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