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Lifting of supersingular points on X0 (pr) and lower bound of ramification index

Published online by Cambridge University Press:  22 January 2016

Fumiyuki Momose  and
Mahoro Shimura
Fumiyuki Momose
Affiliation:
Department of Mathematics, Chuo University, 1-13-27, Kasuga Bunkyou-ku Tokyo, 112, Japan, momose@math.chuo-u.ac.jp
Mahoro Shimura
Affiliation:
Department of Mathematics, Waseda University, 3-4-1, Okubo Shinjuku-ku Tokyo, 169-8555, Japan, shimura@gm.math.waseda.ac.jp
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Abstract

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Let K be a finite extension of (= the maximal unramified extension of Qp) of degree eK, its integer ring, p a rational prime and r a positive integer. If there exists a one parameter formal group defined over whose reduction is of height 2 with a cyclic subgroup V of order pr defined over , then .

We apply this result to a criterion for non-existence of Q-rational point of . (This criterion is Momose’s theorem in [14] except for the cases p = 5 and p = 13, but our new proof does not require defining equations of modular curves except for the case p = 2.)

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 165 , March 2002 , pp. 159 - 178
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

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