A mathematical framework links two questions about abeloid varieties over p-adic fields: how their reduction can be recognized, and how their rational points can be described through conjugate uniformization. The proof-based study brings rigid analytic 1-motives, monodromy pairings, Galois representations and p-divisible rigid analytic groups into a single account.
The objects under study are proper, smooth and connected rigid analytic groups over K, including generalizations of groups associated with abelian varieties. The paper develops categorical equivalences and exact constructions for these objects.
A test for when the geometry improves
One central result concerns the geometric monodromy pairing attached to a rigid analytic 1-motive. In the theorem’s stated setting, the motive has potentially good reduction if and only if that pairing is trivial. The criterion turns a reduction question into a condition on the pairing.
In a second result, a monodromy-based modification of the motive has good reduction exactly when the original motive has semi-stable reduction, according to the theorem. That gives the two reduction behaviors a direct link within the same formalism.
The paper also reports equivalences between formal and log formal 1-motives over O_K and rigid analytic 1-motives over K. The formal category corresponds to good reduction, while the log formal category corresponds to semi-stable reduction.
Reduction read through Galois data
Those reduction questions are also recast in Galois-representation terms. For a rigid analytic 1-motive, good reduction is equivalent to the stated unramified ℓ-adic conditions and the stated crystalline p-adic condition.
Under the stated assumptions, semi-stable reduction is equivalent to the ℓ-adic condition. In characteristic zero, the theorem adds equivalences with the other stated integral and p-adic conditions. The result is conditional on the field, residue-field and representation hypotheses in the theorem.
For a semi-stable abeloid variety, the construction extends it over O_K as a log formal 1-motive and supplies an associated log p-divisible group over O_K.
A bridge to p-adic classification
To handle conjugate uniformization, the analysis uses Fargues’ theory of p-divisible rigid analytic groups and treats rational points through Hodge-Tate triples. It states that dualizable p-divisible rigid analytic groups over K are equivalent to finite-rank Hodge-Tate Zp-representations of ΓK with Hodge-Tate weights 0 and 1.
Each rigid analytic 1-motive also receives a naturally attached dualizable de Rham p-divisible rigid analytic group fitting into the paper’s stated exact sequence. This gives the 1-motive a corresponding p-divisible object within the same framework.
Describing rational points
For an abeloid variety, the authors give a canonical ΓK-equivariant decomposition of its rational points as a topological group. It separates prime-to-p torsion from the p-topologically torsion component while preserving the Galois action.
The integration map receives two precise characterizations. When T_A^ét is zero, its kernel is the subgroup of p-divisible points in A⟨p∞⟩(K). For the associated extension, a point lies in the image exactly when it admits a rigidification.
The claims are conditional
The work is an arXiv preprint, version 1, dated 26 August 2026.
The results should be read within their stated mathematical setting. The reduction criteria depend on the field, residue field and representation assumptions, while the Hodge-Tate classification is stated for the specified p-adic base and weights. The paper’s conclusions are theorem-level equivalences and constructions under those conditions.
The supplied acknowledgements report partial support for the second author from the CAS Project for Young Scientists in Basic Research and the NSFC, and an invitation and hospitality at DPMMS, University of Cambridge, for the third author.
Paper data and sources
Original title: Rigid analytic 1-motives and conjugate uniformization of abeloid varieties
Authors: Khai-Hoan Nguyen-Dang, Xu Shen, Heer Zhao
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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