From the Birch and Swinnerton-Dyer Conjecture to non-commutative Iwasawa theory via the Equivariant Tamagawa Number Conjecture - a survey

Abstract

Introduction This paper aims to give a survey on Fukaya and Kato's article [23] which establishes the relation between the Equivariant Tamagawa Number Conjecture (ETNC) of Burns and Flach [9] and the noncommutative Iwasawa Main Conjecture (MC) (with p -adic L -function) as formulated by Coates, Fukaya, Kato, Sujatha and the author [14]. Moreover, we compare their approach with that of Huber and Kings [24] who formulate an Iwasawa Main Conjecture (without p -adic L -functions). We do not discuss these conjectures in full generality here, in fact we are mainly interested in the case of an abelian variety defined over ℚ. Nevertheless we formulate the conjectures for general motives over ℚ as far as possible. We follow closely the approach of Fukaya and Kato but our notation is sometimes inspired by [9, 24]. In particular, this article does not contain any new result, but hopefully serves as introduction to the original articles. See [47] for a more down to earth introduction to the GL 2 Main Conjecture for an elliptic curve without complex multiplication. There we had pointed out that the Iwasawa main conjecture for an elliptic curve is morally the same as the (refined) Birch and Swinnerton Dyer (BSD) Conjecture for a whole tower of number fields. The work of Fukaya and Kato makes this statement precise as we are going to explain in these notes. For the convenience of the reader we have given some of the proofs here which had been left as an exercise in [23] whenever we had the feeling that the presentation of the material becomes more transparent thereby.