The revised relative Manin–Mumford conjecture in torsion-point form

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Let G/S\mathcal{G}/S be a semi-abelian scheme of relative dimension g+ng+n with toric rank nn, and let SS be an irreducible variety over C\mathbb{C}. Let ZZ be an irreducible closed subvariety of G\mathcal{G}, dominant over SS. Write Gtor\mathcal{G}_{\mathrm{tor}} for the fiberwise torsion points.

Revised relative Manin–Mumford conjecture. If Z(C)∩GtorZ(\mathbb{C})\cap\mathcal{G}_{\mathrm{tor}} is Zariski dense in ZZ, then either ZZ is contained in a special subvariety of G\mathcal{G}, or dim⁡Z≥g+n\dim Z\geq g+n.

The source calls this an immediate corollary of the revised relative conjecture. It is proposed after the original formulation is shown to fail because of Ribet sections.

References

Primary source

Kaiyuan Gu and Chenxin Huang, “On Special Subvarieties of the Universal Semi-abelian Scheme and Pink Conjectures”, arXiv:2410.17755 (2024).

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