Chai's conjectures on base change conductors

TL;DR

This paper studies the base change conductor invariant for semiabelian varieties, proving its additivity in exact sequences, invariance under duality, and providing new proofs and counterexamples related to its properties.

Contribution

It confirms Chai's conjecture on the additivity of the base change conductor and shows that a proposed generalisation does not hold, advancing understanding of this invariant.

Findings

Proves additivity of the base change conductor in short exact sequences.

Shows invariance of the conductor under duality in equal positive characteristic.

Provides a new proof relating the conductor of a torus to its cocharacter module.

Abstract

The base change conductor is an invariant introduced by Chai which measures the failure of a semiabelian variety to have semiabelian reduction. We investigate the behaviour of this invariant in short exact sequences, as well as under duality and isogeny. Our results imply Chai's conjecture on the additivity of the base change conductor in short exact sequences, while also showing that a proposed generalisation of this conjecture fails. We use similar methods to show that the base change conductor is invariant under duality of Abelian varieties in equal positive characteristic (answering a question of Chai), as well as giving a new short proof of a formula due to Chai, Yu, and de Shalit which expresses the base change conductor of a torus in terms of its (rational) cocharacter module.