Prolongement de biextensions et accouplements en cohomologie log plate

TL;DR

This paper investigates extending biextensions of smooth commutative group schemes using log schemes, overcoming obstructions present in classical topologies, and applies this to Néron models and pairings.

Contribution

It demonstrates that biextensions can be extended in the log flat topology, enabling new pairings on Néron models that unify existing pairings by Mazur, Tate, and Grothendieck.

Findings

Extension of biextensions in log flat topology

Extension of Weil biextension to Néron models

Unified pairing combining class group and monodromy

Abstract

We study, using the language of log schemes, the problem of extending biextensions of smooth commutative group schemes by the multiplicative group. This was first considered by Grothendieck in SGA 7. We show that this problem admits a solution in the category of sheaves for Kato's log flat topology, in contradistinction to what can be observed using the fppf topology, for which monodromic obstructions were defined by Grothendieck. In particular, in the case of an abelian variety and its dual, it is possible to extend the Weil biextension to the whole N\'eron model. This allows us to define a pairing on the points which combines the class group pairing defined by Mazur and Tate and Grothendieck's monodromy pairing.