Extension of relative rigid homomorphisms from the formal multiplicative group

TL;DR

This paper extends a known theorem about rigid group homomorphisms from the formal multiplicative group to a broader relative setting over certain rigid spaces, under additional reduction assumptions.

Contribution

It proves a relative version of L"utkebohmert's theorem for rigid group homomorphisms over geometrically reduced quasi-compact spaces, with a new hypothesis on good reduction.

Findings

The theorem holds under the new relative setting with the additional hypothesis.

It facilitates the study of rigid uniformisation of abelian and semiabelian varieties.

Provides a foundation for further research in relative rigid uniformisation.

Abstract

A theorem of L\"utkebohmert states that a rigid group homomorphism from the formal multiplicative group to a smooth commutative rigid group $G$, with relatively compact image, can be extended to a homomorphism from the rigid multiplicative group to $G$. In this paper, we prove a relative version of this theorem over a geometrically reduced quasi-compact quasi-separated rigid space. The relative theorem is proved under an additional hypothesis that some open relative subgroup of $G$ has good reduction. This theorem is useful for studying rigid uniformisation of abelian or semiabelian varieties in a relative setting.