The revised relative Manin–Mumford conjecture for semi-abelian schemes

Let GS\mathcal{G}\to S be a semi-abelian scheme over C\mathbb{C}, and let ZGZ\subset\mathcal{G} be an irreducible closed subvariety of dimension dd. Assume that ZZ is not contained in a proper special subvariety of G\mathcal{G}. Define G[>d]\mathcal{G}^{[>d]} fiberwise as the union of algebraic subgroups of codimension greater than dd.

Revised relative unlikely-intersection conjecture. The intersection ZG[>d]Z\cap\mathcal{G}^{[>d]} is not Zariski dense in ZZ.

Here special subvarieties include the additional subvarieties, such as Ribet sections, identified through the mixed Shimura classification. The source presents this as the revised version of Pink's relative conjecture.

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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