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