Neron models of 1-motives and duality

TL;DR

This paper extends the concept of Neron models to arbitrary Deligne 1-motives over Dedekind schemes, establishing a duality property that generalizes Grothendieck's conjecture, with specific conditions on residue fields and reduction types.

Contribution

It introduces a new definition of Neron models for 1-motives, incorporating ramification and semi-abelian parts, and proves a duality property generalizing Grothendieck's conjecture.

Findings

Neron models defined for all Deligne 1-motives over Dedekind schemes.

Duality property holds under certain conditions on residue fields.

The approach extends Grothendieck's philosophy on motives of arbitrary weights.

Abstract

In this paper, we propose a definition of Neron models of arbitrary Deligne 1-motives over Dedekind schemes, extending Neron models of semi-abelian varieties. The key property of our Neron models is that they satisfy a generalization of Grothendieck's duality conjecture in SGA 7 when the residue fields of the base scheme at closed points are perfect. The assumption on the residue fields is unnecessary for the class of 1-motives with semistable reduction everywhere. In general, this duality holds after inverting the residual characteristics. The definition of Neron models involves careful treatment of ramification of lattice parts and its interaction with semi-abelian parts. This work is a complement to Grothendieck's philosophy on Neron models of motives of arbitrary weights.