On $\psi$-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture
Authors:
Richard Pink and Damian Roessler
Journal:
J. Algebraic Geom. 13 (2004), 771-798
DOI:
https://doi.org/10.1090/S1056-3911-04-00368-6
Published electronically:
February 11, 2004
MathSciNet review:
2073195
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References |
Additional Information
Abstract: Let $A$ be a semiabelian variety over an algebraically closed field of arbitrary characteristic, endowed with a finite morphism $\psi : A\to A$. In this paper, we give an essentially complete classification of all $\psi$-invariant subvarieties of $A$. For example, under some mild assumptions on $(A,\psi )$ we prove that every $\psi$-invariant subvariety is a finite union of translates of semiabelian subvarieties. This result is then used to prove the Manin-Mumford conjecture in arbitrary characteristic and in full generality. Previously, it had been known only for the group of torsion points of order prime to the characteristic of $K$. The proofs involve only algebraic geometry, though scheme theory and some arithmetic arguments cannot be avoided.
Abr Abramovich, D.: Subvarieties of semiabelian varieties. Compositio Math. 90 (1994), 37–52.
BogomolovCR Bogomolov, F. A.: Sur l’algébricité des représentations $\ell$-adiques. C. R. Acad. Sci. Paris, Sér. A, t. 290 (1980), 701–703.
Bo Bogomolov, F. A.: Points of finite order on abelian varieties. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 782–804.
BousBour Bouscaren, E: Théorie des modèles et conjecture de Manin-Mumford [d’après Ehud Hrushovski]. Sém. Bourbaki no. 870 (1999–2000).
Font Fontaine, M.: Groupes $p$-divisibles sur les corps locaux. Astérisque 47–48 (1977).
SGA1 Grothendieck, A., et al.: Revêtements étales et groupe fondamental. Séminaire de Géométrie Algébrique du Bois-Marie 1960–61, (SGA1), Lect. Notes in Math. 224, Berlin etc.: Springer (1971).
SGA3 Grothendieck, A., et al.: Schémas en Groupes I–III. Séminaire de Géométrie Algébrique du Bois-Marie 1962/64 (SGA3), Lect. Notes in Math. 151–153, Berlin etc.: Springer (1970).
HRJAMS Hrushovski, E.: The Mordell-Lang conjecture for function fields. J. Amer. Math. Soc. 9, no. 3 (1996), 667–690.
HR Hrushovski, E.: The Manin-Mumford conjecture and the model theory of difference fields. Ann. Pure Appl. Logic 112 (2001), no. 1, 43–115.
Lang Lang, S.: Algebraic groups over finite fields. Amer. J. Math. 78 (1956), 555–563.
Manin Manin, Yu. I.: The theory of commutative formal groups over fields of finite characteristic. Russian Math. Surveys 18 No. 6 (1963), 1–83.
Oesterle Oesterlé, J.: Courbes sur une variété abélienne [d’après M. Raynaud]. Sém. Bourbaki 625 (1983–84).
PR1 Pink, R., Roessler, D.: On Hrushovski’s proof of the Manin-Mumford conjecture. Preprint May 31, 2002, 6p.
Ra1 Raynaud, M.: Courbes sur une variété abélienne et points de torsion. Invent. Math. 71 (1983), no. 1, 207–233.
Ra2 Raynaud, M.: Sous-variétés d’une variété abélienne et points de torsion. Arithmetic and geometry, Vol. I, 327–352, Progr. Math. 35, Birkhäuser Boston, Boston, MA, 1983.
Serre Serre, J.-P.: Oeuvres, vol. IV (1985-1998). Springer 2000.
Ullmo Ullmo, E.: Positivité et discrétion des points algébriques des courbes. Ann. of Math. (2) 147 (1998), no. 1, 167–179.
Weil Weil, A.: Variétés abéliennes et courbes algébriques. Paris: Hermann, 1948.
Zhang Zhang, S.-W.: Equidistribution of small points on abelian varieties. Ann. of Math. (2) 147 (1998), no. 1, 159–165.
Abr Abramovich, D.: Subvarieties of semiabelian varieties. Compositio Math. 90 (1994), 37–52.
BogomolovCR Bogomolov, F. A.: Sur l’algébricité des représentations $\ell$-adiques. C. R. Acad. Sci. Paris, Sér. A, t. 290 (1980), 701–703.
Bo Bogomolov, F. A.: Points of finite order on abelian varieties. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 782–804.
BousBour Bouscaren, E: Théorie des modèles et conjecture de Manin-Mumford [d’après Ehud Hrushovski]. Sém. Bourbaki no. 870 (1999–2000).
Font Fontaine, M.: Groupes $p$-divisibles sur les corps locaux. Astérisque 47–48 (1977).
SGA1 Grothendieck, A., et al.: Revêtements étales et groupe fondamental. Séminaire de Géométrie Algébrique du Bois-Marie 1960–61, (SGA1), Lect. Notes in Math. 224, Berlin etc.: Springer (1971).
SGA3 Grothendieck, A., et al.: Schémas en Groupes I–III. Séminaire de Géométrie Algébrique du Bois-Marie 1962/64 (SGA3), Lect. Notes in Math. 151–153, Berlin etc.: Springer (1970).
HRJAMS Hrushovski, E.: The Mordell-Lang conjecture for function fields. J. Amer. Math. Soc. 9, no. 3 (1996), 667–690.
HR Hrushovski, E.: The Manin-Mumford conjecture and the model theory of difference fields. Ann. Pure Appl. Logic 112 (2001), no. 1, 43–115.
Lang Lang, S.: Algebraic groups over finite fields. Amer. J. Math. 78 (1956), 555–563.
Manin Manin, Yu. I.: The theory of commutative formal groups over fields of finite characteristic. Russian Math. Surveys 18 No. 6 (1963), 1–83.
Oesterle Oesterlé, J.: Courbes sur une variété abélienne [d’après M. Raynaud]. Sém. Bourbaki 625 (1983–84).
PR1 Pink, R., Roessler, D.: On Hrushovski’s proof of the Manin-Mumford conjecture. Preprint May 31, 2002, 6p.
Ra1 Raynaud, M.: Courbes sur une variété abélienne et points de torsion. Invent. Math. 71 (1983), no. 1, 207–233.
Ra2 Raynaud, M.: Sous-variétés d’une variété abélienne et points de torsion. Arithmetic and geometry, Vol. I, 327–352, Progr. Math. 35, Birkhäuser Boston, Boston, MA, 1983.
Serre Serre, J.-P.: Oeuvres, vol. IV (1985-1998). Springer 2000.
Ullmo Ullmo, E.: Positivité et discrétion des points algébriques des courbes. Ann. of Math. (2) 147 (1998), no. 1, 167–179.
Weil Weil, A.: Variétés abéliennes et courbes algébriques. Paris: Hermann, 1948.
Zhang Zhang, S.-W.: Equidistribution of small points on abelian varieties. Ann. of Math. (2) 147 (1998), no. 1, 159–165.
Additional Information
Richard Pink
Affiliation:
Department of Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland
MR Author ID:
139765
Email:
pink@math.ethz.ch
Damian Roessler
Affiliation:
Department of Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland
Email:
roessler@math.ethz.ch
Received by editor(s):
July 17, 2002
Received by editor(s) in revised form:
September 13, 2002
Published electronically:
February 11, 2004
