Let $k$ be a field, $S/k$ a smooth variety with function field $K$ and $U$ a nonempty open subscheme of $S$. For every finite separable extension $E/K$ we denote by $S^E$ (resp. $U^E$) the normalization of $S$ (resp $U$) in $E$. Let $A/U$ be an abelian scheme with generic fibre $A_\eta$. For every prime number $\ell$ different from $char(k)$ and every finite separable extension $E/K$ we consider the representation $\rho_{\ell, E}$ of $\pi_1(U^E)$ on the $\ell$-torsion part $A_\eta[\ell]$. Let $H(E)$ be the kernel of the epimorphism $\pi_1(U^E)\to \pi_1(S^E)$.

Question: Does there exist a finite separable extension $E/K$ such that for every prime number $\ell\neq char(k)$ the group $\rho_{\ell, E}(H(E))$ is generated by its $\ell$-Sylow subgroups?

Remark: It is known that the answer is yes in the special case $dim(S)=1$. This is a consequence of the semistable reduction theorem of Grothendieck.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.