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

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 dimZg+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.

Sources & referencesView supporting material

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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