Logarithmic Geometry and Geometric Class Field Theory

Abstract

We demonstrate an application of logarithmic geometry in the context of geometric Langlands, by providing a logarithmic upgrade of Deligne's geometric class field theory for tamely ramified Galois groups. 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 logarithmically 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.