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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction
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by Tetsushi Ito PDF
Proc. Amer. Math. Soc. 134 (2006), 2857-2860 Request permission

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
  • Email: tetsushi@math.kyoto-u.ac.jp
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 2857-2860
  • MSC (2000): Primary 14K12; Secondary 11G10, 14G15
  • DOI: https://doi.org/10.1090/S0002-9939-06-08518-2
  • MathSciNet review: 2231608