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Some congruences for non-CM elliptic curves

Research output: Chapter in Book/Report/Conference proceedingConference paper

Original languageEnglish
Title of host publicationElliptic Curves, Modular Forms and Iwasawa Theory
EditorsDavid Loeffler, Sarah Zerbes
PublisherSpringer
Pages295-309
Number of pages15
Volume188
ISBN (Electronic)9783319450322
ISBN (Print)9783319450315
DOIs
E-pub ahead of print16 Jan 2017
EventConference on Elliptic Curves, Modular Forms and Iwasawa Theory, held in honour of the 70th birthday of John H. Coates, 2015 - Cambridge, United Kingdom
Duration: 25 Mar 201527 Mar 2015

Publication series

NameSpringer Proceedings in Mathematics & Statistics
PublisherSpringer
Volume188

Conference

ConferenceConference on Elliptic Curves, Modular Forms and Iwasawa Theory, held in honour of the 70th birthday of John H. Coates, 2015
CountryUnited Kingdom
CityCambridge
Period25/03/201527/03/2015

King's Authors

Abstract

Let p be an odd prime and let G be a p-adic Lie group. The group K1(Λ(G)), for the Iwasawa algebra Λ(G), is well understood in terms of congru- ences between elements of Iwasawa algebras of abelian sub-quotients of G due to the work of Ritter-Weiss and Kato (generalised by the author). In the former one needs to work with all abelian subquotients of G whereas in Kato’s approach one can work with a certain well-chosen sub-class of abelian sub-quotients of G. For instance in [11] K1(Λ(G)) was computed for meta-abelian pro-p groups G but the congruences in this description could only be proved for p-adic L-functions of totally real fields for certain special meta-abelian pro-p groups. By changing the class of abelian subquotients a different description of K1(Λ(G)), for a general G, was obtained in [12] and these congruences were proven for p-adic L-functions of totally real fields in all cases. In this note we propose a strategy to get an alternate description of K1(Λ(G)) when G = GL2(Zp). For this it is sufficient to compute K1 (Zp [GL2 (Z/pn )]). We demonstrate how the strategy should work by explicitly computing K1(Zp[GL1(Z/p)])(p), the pro-p part of K1(Zp[GL2(Z/p)]), which is the most interesting part.

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