O-minimality and certain atypical intersections

By P. Habegger, Jonathan Pila. 2014

101 citations recorded by the source corpus.

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8fed30c3a84805fd6d7a778ceddbe83bfc0c24bf

How researchers see this paper

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In Computational Theory and Mathematics

1. Ziyang Gao
2. P. Habegger
3. V. Dimitrov

In Geometry and Topology

1. Ziyang Gao
2. P. Habegger
3. V. Dimitrov
4. Jacob Tsimerman
5. Benjamin Bakker

In Mathematical Physics

1. Ziyang Gao
2. P. Habegger
3. Jacob Tsimerman
4. Benjamin Bakker

Citations in the corpus

• On the distribution of the Hodge locus (2021, unknown): “Starting from Pila and Zannier [50], a link be-tween functional transcendence and the Zilber–Pink conjecture has been observed in [19, 31, 60], at least for subvarieties of abelian and pure Shimura varieties.”
• Uniformity in Mordell–Lang for curves (2020, background): “In this step the notion of weakly optimal subvarieties introduced by the third-named author and Pila [HP16] is involved.”
• Applications of the hyperbolic Ax–Schanuel conjecture (2017, unknown): “The former, following from another theorem of Ax [3], has recently been used by Habegger and Pila in their proof of the Zilber-Pink conjecture for curves in abelian varieties [18].”