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SELMER GROUPS OF ELLIPTIC CURVES OVER THE $PGL(2)$ EXTENSION

Part of: Arithmetic algebraic geometry Algebraic number theory: global fields

Published online by Cambridge University Press:  30 May 2022

JISHNU RAY Open the ORCID record for JISHNU RAY [Opens in a new window]  and
R. SUJATHA Open the ORCID record for R. SUJATHA [Opens in a new window]
JISHNU RAY
Affiliation:
Institute for Advancing Intelligence TCG Centres for Research and Education in Science and Technology 1st Floor, Tower 1, Bengal Eco Intelligent Park (Techna Building) Block EM, Plot No 3, Sector V, Salt Lake Kolkata 700091, India jishnu.ray@tcgcrest.org, jishnuray1992@gmail.com
R. SUJATHA
Affiliation:
Department of Mathematics The University of British Columbia Room 121, 1984 Mathematics Road Vancouver, BC V6T 1Z2, Canada sujatha@math.ubc.ca
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Abstract

Iwasawa theory of elliptic curves over noncommutative $GL(2)$ extension has been a fruitful area of research. Over such a noncommutative p-adic Lie extension, there exists a structure theorem providing the structure of the dual Selmer groups for elliptic curves in terms of reflexive ideals in the Iwasawa algebra. The central object of this article is to study Iwasawa theory over the $PGL(2)$ extension and connect it with Iwasawa theory over the $GL(2)$ extension, deriving consequences to the structure theorem when the reflexive ideal is the augmentation ideal of the center. We also show how the dual Selmer group over the $GL(2)$ extension being torsion is related with that of the $PGL(2)$ extension.

MSC classification

Primary: 11R23: Iwasawa theory
Secondary: 11G05: Elliptic curves over global fields 11R34: Galois cohomology
Type
Article
Information
Nagoya Mathematical Journal , Volume 248 , December 2022 , pp. 922 - 938
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Copyright
© (2022) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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Footnotes

This work started with the Pacific Institute for the Mathematical Sciences and the Centre National de la Recherche Scientifique research funding received by Jishnu Ray. Later on, he also received funding from the Tata Institute of Fundamental Research and the Institute for Advancing Intelligence, The Chatterjee Group—Centres for Research and Education in Science and Technology in writing the revised versions. R. Sujatha gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada Discovery grant 2019-03987.

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