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On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction

Author(s): Tetsushi Ito
Journal: Proc. Amer. Math. Soc. 134 (2006), 2857-2860.
MSC (2000): Primary 14K12; Secondary 11G10, 14G15
Posted: May 1, 2006
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Abstract: We give a short proof of the `` prime-to-$ p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $ p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $ p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$ p$ torsion points via nontrivial homothety.

References:

[Bog]
F. A. Bogomolov, Sur l'algébricité des représentations $ l$-adiques, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 15, A701-A703. MR 0574307 (81c:14025)

[BZ]
E. Bombieri, U. Zannier, Heights of algebraic points on subvarieties of abelian varieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 4, 779-792. MR 1469574 (98j:11043)

[E1]
N. D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over $ {\bf Q}$, Invent. Math. 89 (1987), no. 3, 561-567. MR 0903384 (88i:11034)

[E2]
N. D. Elkies, Supersingular primes for elliptic curves over real number fields, Compositio Math. 72 (1989), no. 2, 165-172. MR 1030140 (90i:11058)

[Hi]
M. Hindry, Autour d'une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575-603. MR 0969244 (89k:11046)

[Hr]
E. Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43-115. MR 1854232 (2003d:03061)

[L1]
S. Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229-234. MR 0190146 (32:7560)

[L2]
S. Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1970. MR 0282947 (44:181)

[Mi]
J. S. Milne, Abelian varieties, in Arithmetic geometry (Storrs, Conn., 1984), 103-150, Springer, New York, 1986. MR 0861974

[Mu]
D. Mumford, Abelian varieties, Published for the Tata Institute of Fundamental Research, Bombay, 1970. MR 0282985 (44:219)

[LO]
K.-Z. Li, F. Oort, Moduli of supersingular abelian varieties, Lecture Notes in Math., 1680, Springer, Berlin, 1998. MR 1611305 (99e:14052)

[PR1]
R. Pink, D. Roessler, On Hrushovski's proof of the Manin-Mumford conjecture, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 539-546, Higher Ed. Press, Beijing, 2002. MR 1989204 (2004f:14062)

[PR2]
R. Pink, D. Roessler, On $ \psi$-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture, J. Algebraic Geom. 13 (2004), no. 4, 771-798. MR 2073195 (2005d:14061)

[Ra1]
M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), no. 1, 207-233. MR 0688265 (84c:14021)

[Ra2]
M. Raynaud, Sous-variétés d'une variété abélienne et points de torsion, in Arithmetic and geometry, Vol. I, 327-352, Progr. Math., 35, Birkhäuser, Boston, Boston, MA, 1983. MR 0717600 (85k:14022)

[Ro]
D. Roessler, A note on the Manin-Mumford conjecture, preprint, math.NT/0409083, 2004.

[Ta]
J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134-144. MR 0206004 (34:5829)

[Tz]
P. Tzermias, The Manin-Mumford conjecture: a brief survey, Bull. London Math. Soc. 32 (2000), no. 6, 641-652. MR 1781574 (2001g:11091)

[U]
E. Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), no. 1, 167-179. MR 1609514 (99e:14031)

[Z]
S.-W. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159-165. MR 1609518 (99e:14032)

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Additional Information:

Tetsushi Ito
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
Email: tetsushi@math.kyoto-u.ac.jp

DOI: 10.1090/S0002-9939-06-08518-2
PII: S 0002-9939(06)08518-2
Received by editor(s): December 7, 2004
Received by editor(s) in revised form: April 28, 2005
Posted: May 1, 2006
Communicated by: Michael Stillman
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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