Logarithmic Geometry and Geometric Class Field Theory

TL;DR

This paper advances geometric class field theory by integrating logarithmic geometry, establishing a new correspondence between local systems and multiplicative local systems on a logarithmic Picard space, and providing a geometric interpretation of local-global compatibility.

Contribution

It introduces a logarithmic framework for class field theory, defining a framed logarithmic Picard space and demonstrating a bijection with local systems, extending the geometric Langlands correspondence.

Findings

Logarithmic compactification yields log simply connected fibers.

Canonical bijection between local systems and multiplicative local systems.

Re-derivation of tamely ramified global Artin reciprocity.

Abstract

In this paper, we provide an upgrade of Deligne's geometric class field theory for tamely ramified Galois groups using logarithmic geometry. In particular, we define a framed logarithmic Picard space, and show that a logarithmic compactification of the classical tamely ramified Div-to-Pic map has, for sufficiently large degree, log simply connected fibers given by loagrithmically compactified vector spaces. This provides a canonical bijection between local systems on the curve with divisorial log structure and multiplicative local systems on the framed logarithmic Picard (a logarithmic version of the Hecke eigensheaf correspondence of Geometric Langlands for GL_1. We use this to re-derive tamely ramified global Artin reciprocity for function fields, and show that logarithmic geometry allows for a geometric interpretation of local-to-global compatibility at all places (in addition to the unramified places).