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An example of non-cotorsion Selmer group


Authors: King Fai Lai, Ignazio Longhi, Ki-Seng Tan and Fabien Trihan
Journal: Proc. Amer. Math. Soc. 143 (2015), 2355-2364
MSC (2010): Primary 11S40; Secondary 11R23, 11R34, 11R42, 11R58, 11G05, 11G10
Published electronically: January 21, 2015
MathSciNet review: 3326018
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Abstract: Let $ A/K$ be an elliptic curve over a global field of characteristic $ p>0$. We provide an example where the Pontrjagin dual of the Selmer group of $ A$ over a $ \Gamma :=\mathbb{Z}_p$-extension $ L/K$ is not a torsion $ \mathbb{Z}_p[[\Gamma ]]$-module and show that the Iwasawa Main Conjecture for $ A/L$ holds nevertheless.


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Additional Information

King Fai Lai
Affiliation: School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China
Email: kinglaihonkon@gmail.com

Ignazio Longhi
Affiliation: Department of Mathematics, National Taiwan University. Taipei 10764, Taiwan
Address at time of publication: Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, No. 111 Ren’ai Road, Dushu Lake Higher Education Town, Suzhou Industrial Park, Suzhou 215123 Jiangsu, People’s Republic of China.
Email: longhi@math.ntu.edu.tw

Ki-Seng Tan
Affiliation: Department of Mathematics, National Taiwan University, Taipei 10764, Taiwan
Email: tan@math.ntu.edu.tw

Fabien Trihan
Affiliation: College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter, United Kingdom
Address at time of publication: Department of Information and Communication Sciences, Faculty of Science and Technology, Sophia University, 4 Yonbancho, Chiyoda-ku, Tokyo 102-0081 Japan
Email: f-trihan-52m@sophia.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2015-12459-8
Keywords: Abelian variety, Selmer group, Frobenius, Iwasawa theory, Stickelberger element, syntomic
Received by editor(s): August 6, 2013
Received by editor(s) in revised form: January 21, 2014
Published electronically: January 21, 2015
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society