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

TL;DR

This paper investigates the abelian geometric fundamental group of curves over p-adic fields, providing bounds for a subgroup classifying certain ramified coverings, under specific reduction and rational point assumptions.

Contribution

It introduces bounds for a subgroup of the abelian geometric fundamental group related to ramified coverings over the special fiber of the curve.

Findings

Established upper and lower bounds for the subgroup of interest.

Analyzed the impact of good reduction and ordinary Jacobian on the fundamental group.

Connected class field theory to geometric coverings of curves over p-adic fields.

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 $\pi_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$.