Abelian geometric fundamental groups for curves over a $p$-adic field

Abstract

For a curve $X$ over a $p$-adic field $k$, using the class field theory of $X$ due to S. Bloch and S. Saito we study the abelian geometric fundamental group ${\pi }_1^{\mathrm{ab}}(X{)}^{\mathrm{geo}}$ of $X$. In particular, it is investigated a subgroup of ${\pi }_1^{\mathrm{ab}}(X{)}^{\mathrm{geo}}$ which classifies the geometric and abelian coverings of $X$ which allow possible ramification over the special fiber of the model of $X$. Under the assumptions that $X$ has a $k$-rational point, $X$ has good reduction and its Jacobian variety has good ordinary reduction, we give some upper and lower bounds of this subgroup of ${\pi }_1^{\mathrm{ab}}(X{)}^{\mathrm{geo}}$.