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Pontryagin duality for Iwasawa modules and abelian varieties


Authors: King Fai Lai, Ignazio Longhi, Ki-Seng Tan and Fabien Trihan
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11S40; Secondary 11R23, 11R34, 11R42, 11R58, 11G05, 11G10
DOI: https://doi.org/10.1090/tran/7016
Published electronically: August 15, 2017
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Abstract: We prove a functional equation for two projective systems of finite abelian $ p$-groups, $ \{\mathfrak{a}_n\}$ and $ \{\mathfrak{b}_n\}$, endowed with an action of $ \mathbb{Z}_p^d$ such that $ \mathfrak{a}_n$ can be identified with the Pontryagin dual of $ \mathfrak{b}_n$ for all $ n$.

Let $ K$ be a global field. Let $ L$ be a $ \mathbb{Z}_p^d$-extension of $ K$ ($ d\geq 1$), unramified outside a finite set of places. Let $ A$ be an abelian variety over $ K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $ A$.


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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 Mathematical Sciences, Xi’an Jiaotong-Liverpool University, No.111 Ren’ai Road, Suzhou Dushu Lake Higher Education Town, Suzhou Industrial Park, Jiangsu, People’s Republic of China
Email: Ignazio.Longhi@xjtlu.edu.cn

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

Fabien Trihan
Affiliation: 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/tran/7016
Keywords: Pontryagin duality, abelian variety, Selmer group, Iwasawa theory
Received by editor(s): February 4, 2016
Received by editor(s) in revised form: June 27, 2016
Published electronically: August 15, 2017
Additional Notes: The first, second, and third authors were partially supported by the National Science Council of Taiwan, grants NSC98-2115-M-110-008-MY2, NSC100-2811-M-002-079, and NSC99-2115-M-002-002-MY3, respectively
The fourth author was supported by EPSRC
Article copyright: © Copyright 2017 American Mathematical Society