Abstract:
I will describe the proof of the following result:
Theorem.
Assume that $K=\bar F_p(X),L=\bar F_l(Y)$
where $X,Y$ are algebraic varieties over
$\bar F_p, \bar F_l$ respectively.
Let
$\psi : K^*/\bar F_p^*\to L^*/\bar F_l$ be
homormorphism between the quotients of multiplicative groups.
We make the following assumptions:
1) $ dim X \geq 2, dim Y \geq 2$
2)The images of algebraically dependent elements
are algebraically dependent.
3) There are at least two elements $x,y\in K^*/ \bar F_p$ such that
$\psi(x),\psi(y)$ are algebraically independent
4) $\psi$ has a nontrivial kernel.
Then there is nonarhemedean valuation $\nu$ on $K$ such
that the map $\psi$ on the unit subgroup $A_{\nu}^*$ is obtained
from a composition of natural projection of the valuation ring
$A_{\nu}\to K_{\nu}$ where $K_{\nu}$ is residue field of $\nu$
and a monomorphism $K_{\nu}^*/\bar F_p\to L^*/\bar F_l$.
I will also explain the relation of this result
to abelian version of Grothendieck section conjecture.
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