Some congruences for non-CM elliptic curves

TY - CHAP

T1 - Some congruences for non-CM elliptic curves

AU - Kakde, Mahesh

PY - 2017/1/16

Y1 - 2017/1/16

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85025139538&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-45032-2_8

DO - 10.1007/978-3-319-45032-2_8

M3 - Conference paper

SN - 9783319450315

VL - 188

T3 - Springer Proceedings in Mathematics & Statistics

SP - 295

EP - 309

BT - Elliptic Curves, Modular Forms and Iwasawa Theory

A2 - Loeffler, David

A2 - Zerbes, Sarah

PB - Springer

T2 - Conference on Elliptic Curves, Modular Forms and Iwasawa Theory, held in honour of the 70th birthday of John H. Coates, 2015

Y2 - 25 March 2015 through 27 March 2015

ER -