Hybrid Conjecture in a Mixed Shimura variety
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915958601285632 |
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| author | Richard, Rodolphe Yafaev, Andrei |
| author_facet | Richard, Rodolphe Yafaev, Andrei |
| contents | The authors previously formulated the hybrid conjecture, unifying André-Pink-Zannier and André-Oort conjectures, and proved it in Shimura varieties of abelian type. We study its analogue for mixed Shimura varieties, and consider the prime example, the universal abelian scheme $\mathcal{A}_g\to \mathbb{A}_g$.
In a radical departure from the Pila-Zannier strategy, typically applied to such questions, we employ instead a combination of equidistribution and o-minimality
Our main result strictly includes the following: the Hybrid Conjecture, in particular the André-Pink-Zannier and André-Oort conjectures, for $\mathbb{A}_g$; the mixed André-Oort conjecture for $\mathcal{A}_g$; and Manin-Mumford conjecture for arbitrary abelian varieties. It also yields an analogue of the ``Manin-Mumford in arithmetic pencil", a result of Baldi-Richard-Ullmo, for abelian schemes over a variety.
The mixed hybrid conjecture in $\mathcal{A}_g$ also encompasses the Mordell-Lang conjecture. We actually reduce the mixed hybrid conjecture for $\mathbb{A}_g$ to its "mordellic" part.
We also prove, Galois-theoretic results: uniform variants on the Ribet's Kummer theory of Abelian varieties, and Serre's theorem on Lang's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23376 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hybrid Conjecture in a Mixed Shimura variety Richard, Rodolphe Yafaev, Andrei Number Theory Algebraic Geometry The authors previously formulated the hybrid conjecture, unifying André-Pink-Zannier and André-Oort conjectures, and proved it in Shimura varieties of abelian type. We study its analogue for mixed Shimura varieties, and consider the prime example, the universal abelian scheme $\mathcal{A}_g\to \mathbb{A}_g$. In a radical departure from the Pila-Zannier strategy, typically applied to such questions, we employ instead a combination of equidistribution and o-minimality Our main result strictly includes the following: the Hybrid Conjecture, in particular the André-Pink-Zannier and André-Oort conjectures, for $\mathbb{A}_g$; the mixed André-Oort conjecture for $\mathcal{A}_g$; and Manin-Mumford conjecture for arbitrary abelian varieties. It also yields an analogue of the ``Manin-Mumford in arithmetic pencil", a result of Baldi-Richard-Ullmo, for abelian schemes over a variety. The mixed hybrid conjecture in $\mathcal{A}_g$ also encompasses the Mordell-Lang conjecture. We actually reduce the mixed hybrid conjecture for $\mathbb{A}_g$ to its "mordellic" part. We also prove, Galois-theoretic results: uniform variants on the Ribet's Kummer theory of Abelian varieties, and Serre's theorem on Lang's conjecture. |
| title | Hybrid Conjecture in a Mixed Shimura variety |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2604.23376 |