Let pi:A→ S be an abelian scheme over an algebraically closed field of characteristic 0, and let X⊆A be an irreducible closed subvariety. If the set x∈ X: x is torsion in the fiber A_pi(x) is Zariski dense in X, then X is a special subvariety, namely a relative torsion coset: an irreducible translate of an abelian subscheme by a torsion section, in the sense of Pink and Zannier.

Relative Manin–Mumford Conjecture

The conjecture, proposed by Pink in 2005 and Zannier in 2012, predicts strong restrictions on subvarieties of abelian schemes containing a Zariski-dense set of torsion points.

Known results

Masser and Zannier proved the conjecture for abelian surfaces over the algebraic numbers.
Bertrand, Masser, Pillay, and Zannier (2013) identified Ribet sections as the only obstruction for one-dimensional families of semi-abelian surfaces.
Earlier work treated curves, surfaces, and fibered products of elliptic-curve families.

2026 journal publication

Ziyang Gao and Philipp Habegger’s publication in Inventiones mathematicae reports a proof in characteristic zero. It also gives a torsion-density criterion, namely $\operatorname{rank}_{\mathrm{Betti}}(X)=2g$ under the stated density hypothesis, and a new proof of the uniform curve case. The result is reported as complete, but independent verification is not documented in the retrieved sources.

Current status (as of August 2026): The conjecture is claimed solved in characteristic zero by Gao and Habegger, but the reported proof remains unverified here.

Relative Manin–Mumford Conjecture

Primary source

Additional references

Progress summary

Solutions 0