Rigid analytic 1-motives and conjugate uniformization of abeloid varieties

@misc{nguyendang2026rigid,
  title = {{Rigid analytic 1-motives and conjugate uniformization of abeloid varieties}},
  author = {Khai-Hoan Nguyen-Dang and Xu Shen and Heer Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.25424},
  eprint = {2608.25424},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.25424},
}

TY  - UNPB
TI  - Rigid analytic 1-motives and conjugate uniformization of abeloid varieties
AU  - Khai-Hoan Nguyen-Dang
AU  - Xu Shen
AU  - Heer Zhao
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.25424
UR  - https://arxiv.org/abs/2608.25424
AB  - Let K be a p-adic field. We study the arithmetic theory of abeloid varieties over K. Our aims are twofold. First, we study the theory of rigid analytic 1-motives, which will be viewed as a tool to describe degeneration of abeloid varieties, similarly as in the classical algebraic setting. Our key new results are the equivalence between formal (resp. log formal) 1-motives over 𝒪_K and rigid analytic 1-motives with good (resp. semi-stable) reduction over K, and the Néron-Ogg-Shafarevich criterion for the good (resp. semi-stable) reduction of rigid analytic 1-motives. In particular, we construct log formal 1-motives and log p-divisible groups over 𝒪_K from semi-stable abeloid varieties over K. Next, we study the conjugate uniformization of an arbitrary abeloid variety A over K. This is a type of p-adic uniformization initiated by Iovita--Morrow--Zaharescu in case of abelian varieties with good reduction. Our approach here is based on Fargues' theory of p-divisible rigid analytic groups. In fact, we view the theory of conjugate uniformization as a study of rational points of dualizable p-divisible rigid analytic groups in terms of their classification Hodge--Tate triples. Along the way, we construct p-divisible rigid analytic groups from rigid analytic 1-motives.
ER  -