# Thursday 27 Sep 2018: NT 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.