Some Congruences for Non-CM Elliptic Curves

Abstract

Let p be an odd prime and let G be a p-adic Lie group. The group $K_1(\Lambda (G))$, for the Iwasawa algebra $\Lambda (G)$, is well understood in terms of congruences 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] $K_1(\Lambda (G))$ 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 $K_1(\Lambda (G))$, 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 $K_1(\Lambda (G))$ when $G = GL_2(\mathbb {Z}_p)$. For this it is sufficient to compute $K_1(\mathbb {Z}_p[GL_2(\mathbb {Z}/p^n)])$. We demonstrate how the strategy should work by explicitly computing $K_1(\mathbb {Z}_p[GL_1(\mathbb {Z}/p)])_{(p)}$, the pro-p part of $K_1(\mathbb {Z}_p[GL_2(\mathbb {Z}/p)])$, which is the most interesting part.

(Sant Tukaram)

To John Coates on his 70th birthday.

The author is supported by EPSRC First Grant EP/L021986/1.