Some unlikely intersections beyond André–Oort

DOI10.1112/S0010437X11005604zbMath1288.11062OpenAlexW2077243722MaRDI QIDQ3114531

Jonathan Pila, Philipp Habegger

Publication date: 18 February 2012

Published in: Compositio Mathematica (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1112/s0010437x11005604

zbMATH Keywords

André-Oort conjectureunlikely intersectionZilber-Pink conjecture

Mathematics Subject Classification ID

Complex multiplication and moduli of abelian varieties (11G15) Arithmetic aspects of modular and Shimura varieties (11G18) Heights (11G50) Modular and Shimura varieties (14G35) Complex multiplication and abelian varieties (14K22) Model theory of ordered structures; o-minimality (03C64)

Related Items (22)

Unlikely Intersections with ExCM curves in A2 ⋮ Independence of CM points in elliptic curves ⋮ Height bounds and the Siegel property ⋮ SPECIAL POINT PROBLEMS WITH ELLIPTIC MODULAR SURFACES ⋮ Unlikely intersections with Hecke translates of a special subvariety ⋮ Towards the André-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points ⋮ The relative Bogomolov conjecture for fibered products of elliptic curves ⋮ Hodge theory, between algebraicity and transcendence ⋮ A note on unlikely intersections in Shimura varieties ⋮ Lattices with skew-Hermitian forms over division algebras and unlikely intersections ⋮ Diophantine approximations. Abstracts from the workshop held April 17--23, 2022 ⋮ ENLARGED MIXED SHIMURA VARIETIES, BI-ALGEBRAIC SYSTEM AND SOME AX TYPE TRANSCENDENTAL RESULTS ⋮ Ax-Lindemann for \(\mathcal{A}_g\) ⋮ A Modular André–Oort Statement with Derivatives ⋮ Comparison of volumes of Siegel sets and fundamental domains for \(\mathrm {SL}_n ({\mathbb {Z}})\) ⋮ Multiplicative independence of modular functions ⋮ On a modular Fermat equation ⋮ Rational points on Grassmannians and unlikely intersections in tori ⋮ Some remarks on atypical intersections ⋮ Height estimates for Bianchi groups ⋮ Unlikely intersections with isogeny orbits in a product of elliptic schemes ⋮ Applications of the hyperbolic Ax–Schanuel conjecture

Cites Work

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• On the real exponential field with restricted analytic functions
• The elementary theory of restricted analytic fields with exponentiation
• Heights of CM points on complex affine curves
• Special points on products of modular curves
• Uniform definability of the Weierstrass \(\wp\) functions and generalized tori of dimension one
• Geometric categories and o-minimal structures
• \(ABC\) implies no ``Siegel zeros for \(L\)-functions of characters with negative discriminant
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• The rational points of a definable set
• Projections of semi-analytic sets
• Théorie de Hodge. II. (Hodge theory. II)
• On Schanuel's conjectures
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• EXPONENTIAL SUMS EQUATIONS AND THE SCHANUEL CONJECTURE
• On unlikely intersections of complex varieties with tori
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• On the Hodge groups of some abelian varieties
• Special points on the product of two modular curves
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• Algebraic Groups and Algebraic Dependence
• Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

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