On Integral Class field theory for varieties over $p$-adic fields

TL;DR

This paper establishes a new integral class field theory for varieties over p-adic fields, connecting a novel cohomology theory with the abelianized fundamental group under certain conditions.

Contribution

It introduces a reciprocity isomorphism linking a new cohomology group to an integral model of the abelianized fundamental group for varieties over p-adic fields.

Findings

Proves an isomorphism between a new cohomology group and an integral fundamental group model.

Shows the map becomes an isomorphism of finitely generated abelian groups after base field contribution is removed.

Provides conditions under which the reciprocity law holds for regular, proper, flat varieties over p-adic integers.

Abstract

Let $K$ be a finite extension of the $p$-adic numbers $\mathbb Q_p$ with ring of integers $\mathcal O_K$, $\mathcal X$ a regular scheme, proper, flat, and geometrically irreducible over $\mathcal O_K$ of dimension $d$, and $\mathcal X_K$ its generic fiber. We show, under some assumptions on $\mathcal X_K$, that there is a reciprocity isomorphism of locally compact groups $H_{ar}^{2d-1}(\mathcal X_K, \mathbb Z(d)) \simeq \pi_1^{ab}(\mathcal X_K)_{W}$ from a new cohomology theory to an integral model $\pi_1^{ab}(\mathcal X_K)_{W}$ of the abelianized geometric fundamental groups $\pi_1^{ab}(\mathcal X_K)^{geo}$. After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups.