O-minimality and certain atypical intersections (1409.0771v1)

Abstract: 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.

• The paper verifies the Zilber-Pink Conjecture by reducing it to core arithmetic and functional transcendence statements in modular curves and abelian varieties.
• The paper introduces the large Galois orbit (LGO) hypothesis to establish finiteness results for atypical intersections using o-minimality and point-counting strategies.
• The paper advances diophantine geometry by linking complex conjectures through rigorous reduction methods, paving the way for further exploration in arithmetic geometry.

Overview of "O-Minimality and Certain Atypical Intersections"

The paper, authored by P. Habegger and J. Pila, discusses the application of the o-minimality and point-counting strategies to general problems of "unlikely intersection" type, specifically in the context of the Zilber-Pink Conjecture. This work extends beyond the traditional conjectures like Manin-Mumford and André-Oort, to more generalized settings including products of modular curves and abelian varieties.

Main Contributions

1. Verification of Zilber-Pink Conjecture: The authors verify the Zilber-Pink Conjecture within the framework of modular curves, predicated on the assumptions of lower bounds for Galois orbits and the modular Ax-Schanuel Conjecture. This verification is extended to abelian varieties under specific conditions.
2. Reduction to Core Statements: The research reduces the Zilber-Pink Conjecture to core components: an arithmetic statement and a functional transcendence statement. Notably, the arithmetic statement remains conjectural, indicating areas for further exploration and verification in both modular and abelian contexts.
3. Large Galois Orbit Hypothesis (LGO): The paper introduces the LGO hypothesis, asserting that isolated intersection points in specific subvarieties have large Galois orbits, characterized by a complexity measure tailored to the special subvarieties.
4. Finiteness Results: The authors establish that certain heuristic strategies, notably employing o-minimality, are adequate to handle "atypical intersections" provided additional theoretical components are present. They exemplify this through the requisite finiteness results for geodesic-optimal subvarieties in the context of the LGO hypothesis.
5. Application and Implications: The paper extends applications to certain modular and non-compact settings, projecting potential proofs for broader conjectures under specific conditions. A significant portion is dedicated to verifying that these strategies can, indeed, address complexities introduced by Zilber-Pink through a host of mathematical constructs spanning both modular functions and complex abelian varieties.

Implications and Future Directions

The verifications and reductions achieved in this research mark a substantial theoretical advancement, shedding light on complex interconnections between various conjectures in diophantine geometry. Particularly, the reduction of the Zilber-Pink Conjecture to simpler arithmetic and transcendental problems paves the way for deeper investigations into the foundational elements of arithmetic geometry.

The practical implications extend into understanding how large Galois orbits and their intersection behaviors with special subvarieties offer insights into structural properties in algebraic and transcendental settings. The potential theoretical implications of these findings provide a scaffold for tackling prevailing conjectures, indicating promising directions for future research.

Moreover, this work underscores the utility of o-minimal structures within conjectural landscapes, reaffirming their role as vital tools within modern mathematical methodologies, especially at the intersection of algebraic geometry and diophantine approximation.

Overall, Habegger and Pila's work enhances the comprehension of atypical intersections by entrenching their existence within a broader framework of definable sets and transcends traditional bounds stipulated by foundational conjectures, envisioning new pathways in mathematical research concerning curves and varieties.