ACFA and the Manin-Mumford Conjecture

  • Anand Pillay
Part of the NATO ASI Series book series (ASIC, volume 496)


We give Hrushovski’s proof of the (extended) Manin-Mumford conjecture over number fields. Recall that the Manin-Mumford conjecture states that if X is a nonsingular projective curve of genus > 1 over an algebraically closed field of characteristic 0, and A is the Jacobian variety of X then X ∩ Tor (A) is finite (where Tor (A) denotes the group of torsion points of A). This is of course a special case of the Mordell-Lang conjecture. Raynaud [8] proved that if A is an abelian variety (in characteristic 0) and X is a subvariety of A then X ∩ Tor(A) is a finite union of cosets (proving in particular Manin-Mumford). Hindry [3] proved the same thing, but with A an arbitrary commutative algebraic group.


Irreducible Component Abelian Variety Number Field Finite Union Residue Field 
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  1. 1.
    Z. Chatzidakis, Groups definable in ACFA, this volume.Google Scholar
  2. 2.
    Z. Chatzidakis and E. Hrushovski, Model Theory of difference flelds, preprint 1996.Google Scholar
  3. 3.
    M. Hindry, Points de torsion sur les sousvariétés des variétés abeliennes, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 311–314.MathSciNetzbMATHGoogle Scholar
  4. 4.
    E. Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, preprint 1996.Google Scholar
  5. 5.
    E. Hrushovski and A. Pillay, Weakly normal groups, Logic Colloquium ’85, North Holland, Amsterdam, 1987, pp. 233–244.Google Scholar
  6. 6.
    B. Kim and A. Pillay, Simple theories, to appear in the Annals of Pure and Applied Logic.Google Scholar
  7. 7.
    Y. Kawamata, On Bloch’s conjecture, Invent. Math., 57 (1980), 97–100.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    M. Raynaud, Around the Mordell conjecture for function fields and a conjecture of Serge Lang, Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., 1016, Springer, Berlin-New York, 1983, pp. 1–19.Google Scholar
  9. 9.
    J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, Springer-Verlag, New York-Berlin, 1986.zbMATHCrossRefGoogle Scholar
  10. 10.
    A. Weil, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Anand Pillay
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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