Relative Manin–Mumford for Semi-Abelian Surfaces

DOI10.1017/S0013091515000486zbMath1408.14139arXiv1307.1008MaRDI QIDQ2976316

Umberto Zannier, Daniel Bertrand, Anand Pillay, David W. Masser

Publication date: 28 April 2017

Published in: Proceedings of the Edinburgh Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1307.1008

zbMATH Keywords

differential Galois theoryManin-Mumford conjectureAndré-Oort conjectureZilber-Pink conjecturepolynomial Pell equationssemi-abelian varieties

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Analytic theory of abelian varieties; abelian integrals and differentials (14K20) Modular and Shimura varieties (14G35) Arithmetic ground fields for abelian varieties (14K15) Differential algebra (12H05) Results involving abelian varieties (11J95)

Related Items (18)

Betti maps, Pell equations in polynomials and almost-Belyi maps ⋮ Unlikely intersections in Poincaré biextensions over elliptic schemes ⋮ Cyclotomic torsion points in elliptic schemes ⋮ Towards the André-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points ⋮ Generalized Jacobians and Pellian polynomials ⋮ Pink's conjecture on unlikely intersections and families of semi-abelian varieties ⋮ Unlikely Intersections and Pell’s Equations in Polynomials ⋮ Points de petite hauteur sur une variété semi‐abélienne de la forme Gmn×A$\mathbb {G}_m^n \times A$ ⋮ An overview on problems of unlikely intersections in families of abelian varieties ⋮ Greatest common divisor results on semiabelian varieties and a conjecture of Silverman ⋮ ENLARGED MIXED SHIMURA VARIETIES, BI-ALGEBRAIC SYSTEM AND SOME AX TYPE TRANSCENDENTAL RESULTS ⋮ Torsion points on families of products of elliptic curves ⋮ Pfaffian definitions of Weierstrass elliptic functions ⋮ Relative Manin–Mumford in additive extensions ⋮ Density of algebraic points on Noetherian varieties ⋮ Effective Pila-Wilkie bounds for unrestricted Pfaffian surfaces ⋮ Generic rank of Betti map and unlikely intersections ⋮ Unlikely intersections in semiabelian surfaces

Cites Work

• Unlikely intersections in Poincaré biextensions over elliptic schemes
• \(o\)-minimality and the André-Oort conjecture for \(\mathbb{C}^{n}\)
• Twists and generalized Zolotarev polynomials.
• Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)
• Théories de Galois différentielles et transcendance
• Autour d'une conjecture de Serge Lang. (Around a conjecture by Serge Lang)
• Deficient points on extensions of abelian varieties by \(G_ m\)
• Elliptic functions and transcendence
• Internal DLA and the Gaussian free field
• Ax-Lindemann for \(\mathcal{A}_g\)
• Torsion points on families of products of elliptic curves
• Théorie de Hodge. III
• Rational points on a subanalytic surface.
• A Combination of the Conjectures of Mordell-Lang and André-Oort
• Some Topics in Differential Algebraic Geometry I: Analytic Subgroups of Algebraic Groups

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