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

Author: Tetsushi Ito
Journal: Proc. Amer. Math. Soc. 134 (2006), 2857-2860
MSC (2000): Primary 14K12; Secondary 11G10, 14G15
Published electronically: May 1, 2006
MathSciNet review: 2231608
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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.

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Additional Information

Tetsushi Ito
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

Received by editor(s): December 7, 2004
Received by editor(s) in revised form: April 28, 2005
Published electronically: May 1, 2006
Communicated by: Michael Stillman
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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