On descent theory and main conjectures in non-commutative Iwasawa theory

Abstract

Abstract. We prove a 'Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of noncommutative Iwasawa algebras. We use these results to describe the precise connection between the main conjecture of non-commutative Iwasawa theory (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture.

Algebraic $p$-Adic $L$-Functions in Non-Commutative Iwasawa Theory

“…This construction refines both the notion of 'characteristic element' introduced by Venjakob in [10] and the 'Akashi series' introduced by Coates, Schneider and Sujatha in [5] and also plays a key role in descent theory in non-commutative Iwasawa theory. Indeed, in joint work with Venjakob [3], the results proved here are used to establish a general descent theory that, for example, clarifies the precise connection between main conjectures of non-commutative Iwasawa theory in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob [4] and the relevant cases of the equivariant Tamagawa number conjecture of Flach and the present author [2].…”

“…In joint work with Venjakob [3] we reinterpret Definition 2.2 in terms of the localized K 1 -groups introduced by Fukaya and Kato in [6]. If G has no element of order p, then in [3] we also extend Definition 2.2 (and the results proved in §3 and §4 below) to the case of complexes…”

Iwasawa Theory for One-Parameter Families of Motives

“…Unfortunately, the result is not quite as strong, but it will be sufficient for what follows. 3. Let (T, T 0 ) be a pair of representations associated to a family such that Condition 3.16 on the ramification is satisfied.…”

“…Compare for the theory of generalize Iwasawa invariant also the third section of [3] where in Burns and Venjakob develop a theory for the µ-invariant.…”

On completely faithful Selmer groups of elliptic curves and Hida deformations

“…Inspired by the cyclotomic situation, it was then hoped that such an investigation might give some insight to the noncommutative p-adic L-function which is, even today, still largely conjectural in most situations (although one now has a slightly better understanding of the shape of the p-adic Lfunctions and the form of the main conjecture via an algebraic K-theoretical approach; see [3,4,8,21]). As it turns out, such an approach via global annihilators had been shown to be not feasible in general.…”