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arXiv · 2201.05982

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

Evangelia Gazaki·Toshiro Hiranouchi
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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 $\pi_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$.

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Evangelia Gazaki, Toshiro Hiranouchi. 2022-01-16. Abelian geometric fundamental groups for curves over a $p$-adic field. https://arxiv.org/abs/2201.05982

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