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Some unlikely intersections beyond André–Oort

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Model theory Abelian varieties and schemes

Published online by Cambridge University Press:  30 August 2011

P. Habegger  and
J. Pila
P. Habegger
Affiliation:
Institut fuer Mathematik, Goethe Universitaet Frankfurt, Robert-Mayer-Strasse 6-8, 60325 Frankfurt am Main, Germany (email: habegger@math.uni-frankfurt.ch)
J. Pila
Affiliation:
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK (email: pila@maths.ox.ac.uk)
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Abstract

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According to the André–Oort conjecture, an algebraic curve in Y (1)n that is not equal to a special subvariety contains only finitely many points which correspond to ann-tuple of elliptic curves with complex multiplication. Pink’s conjecture generalizes the André–Oort conjecture to the extent that if the curve is not contained in a special subvariety of positive codimension, then it is expected to meet the union of all special subvarieties of codimension two in only finitely many points. We prove this for a large class of curves in Y (1)n. When restricting to special subvarieties of codimension two that are not strongly special we obtain finiteness for all curves defined over . Finally, we formulate and prove a variant of the Mordell–Lang conjecture for subvarieties of Y (1)n.

Keywords

André–Oort conjectureZilber–Pink conjectureunlikely intersection

MSC classification

Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 03C64: Model theory of ordered structures; o-minimality 11G15: Complex multiplication and moduli of abelian varieties 11G50: Heights 14G35: Modular and Shimura varieties 14K22: Complex multiplication
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 1 - 27
Copyright
Copyright © Foundation Compositio Mathematica 2011

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