The abelian fundamental group with modulus in mixed characteristic

TL;DR

The paper addresses extending Kerz–Saito's abelian fundamental group with modulus to mixed characteristic by defining $\pi_1^{\mathrm{ab}}(X,D)$ for regular flat schemes over a DVR and establishing two equivalent ramification-bounding formulations using local invariants $sw$ and $dt$ alongside global data such as $Sw(\chi)$, $Dt(\chi)$, and the Frobenius–Witt–differential framework. It proves a Lefschetz-type theorem: for a strictly semi-stable model $X$ over $\mathcal{O}_K$ with a suitably ample hypersurface $Y$, the restriction map $\pi_1^{\mathrm{ab}}(Y,E) \to \pi_1^{\mathrm{ab}}(X,D)$ is an isomorphism when $\dim X_K \ge 3$ and a surjection when $\dim X_K=2$, where $E=Y\times_X D$. The result relies on bounding wild ramification via both classical ramification theory and Abbes–Saito, leveraging the Frobenius–Witt differential approach to formulate ampleness and the Swan conductor bounds to pass from the logarithmic to the non-logarithmic setting. This work extends Kerz–Saito to mixed characteristic, connecting ramification theory with differential techniques and providing tools to study wild ramification in a broader arithmetic-geometric context.

Abstract

We define the abelian fundamental group with modulus of a regular flat scheme over a discrete valuation ring, taking into account wild ramification along a divisor. Our definition provides a mixed-characteristic analogue of the abelian fundamental group with modulus introduced by Kerz--Saito for smooth schemes over a perfect field. In this setting, we prove a Lefschetz-type theorem for strictly semi-stable schemes: restriction to a hypersurface of sufficiently large degree relative to the ramification induces an isomorphism of the abelian fundamental groups.

Theorems & Definitions (38)

• Theorem 1: KS14
• Theorem 2: Theorem \ref{['main']}
• Definition 1.1: Ka89
• Proposition 1.2
• proof
• Lemma 1.3
• proof
• Lemma 1.4
• Remark 1.5
• proof