
Contents
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3.1 A method for the Manin-Mumford conjecture 3.1 A method for the Manin-Mumford conjecture
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3.2 Masser’s questions on elliptic pencils 3.2 Masser’s questions on elliptic pencils
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Some remarks on the Question Some remarks on the Question
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3.3 A finiteness proof 3.3 A finiteness proof
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3.4 Related problems, conjectures, and developments 3.4 Related problems, conjectures, and developments
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3.4.1 Pink’s and related conjectures 3.4.1 Pink’s and related conjectures
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3.4.2 Extending Theorem 3.3 from
to C 3.4.2 Extending Theorem 3.3 from
to C
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3.4.3 Effectivity 3.4.3 Effectivity
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3.4.4 Extending Theorem 3.3 to arbitrary pairs of points on families of elliptic curves 3.4.4 Extending Theorem 3.3 to arbitrary pairs of points on families of elliptic curves
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3.4.5 Simple abelian surfaces and Pell’s equations over function fields 3.4.5 Simple abelian surfaces and Pell’s equations over function fields
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3.4.6 Further extensions and analogues 3.4.6 Further extensions and analogues
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3.4.7 Dynamical analogues 3.4.7 Dynamical analogues
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Notes to Chapter 3 Notes to Chapter 3
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1. Torsion values for a single point: Other arguments 1. Torsion values for a single point: Other arguments
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2. A variation on the Manin-Mumford conjecture 2. A variation on the Manin-Mumford conjecture
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Comments on the Methods Comments on the Methods
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3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser
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Published:March 2012
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Abstract
This chapter turns from the multiplicative-group context to the context of abelian varieties. There are here entirely similar results and conjectures: we have already recalled the Manin–Mumford conjecture, and pointed out that the Zilber conjecture also admits an abelian exact analogue. Actually, abelian varieties have moduli, which introduce new issues with respect to the toric case. The chapter focuses mainly on some new problems, raised by Masser, which represent a relative case of Manin–Mumford–Raynaud, where the relevant abelian variety is no longer fixed but moves in a family. The unlikely intersections of Masser's questions occur in the special case of elliptic surfaces (i.e., families of elliptic curves), and can be dealt with by a method that has been recently introduced.
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