Andreas Nickel

On the equivariant Tamagawa number conjecture in tame CM-extensions, II

Let L/K be a finite Galois CM-extension with Galois group G. The Equivariant Tamagawa Number Conjecture (ETNC) for the pair \({(h^0({\rm Spec} (L))(0), {\mathbb Z}G)}\) naturally decomposes into p-parts, where p runs over all rational primes. If p is odd, these p-parts in turn decompose into a plus and a minus part. Let L/K be tame above p. We show that a certain ray class group of L defines an element in \({K_0({\mathbb Z}{_p}G_-, \mathbb Q_p)}\) which is determined by a corresponding Stickelberger element if and only if the minus part of the ETNC at p holds. For this we use the Lifted Root Number Conjecture for small sets of places which is equivalent to the ETNC in the number field case. For abelian G, we show that the minus part of the ETNC at p implies the Strong Brumer–Stark Conjecture at p. We prove the minus part of the ETNC at p for almost all primes p.

Equivariant Iwasawa theory and non-abelian Stark-type conjectures

We discuss three di�erent formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real �elds with Galois group G, where k is a number �eld and G is a p-adic Lie group of dimension 1 for an odd prime p. All these formulations are equivalent and hold if Iwasawas μ-invariant vanishes. Under mild hypotheses, we use this to prove non-abelian generalizations of Brumers conjecture, the Brumer-Stark conjecture and a strong version of the Coates-Sinnott conjecture provided that μ = 0.

Leading terms of Artin L-series at negative integers and annihilation of higher K-groups

Let L / K be a finite Galois extension of number fields with Galois group G . We use leading terms of Artin L -series at strictly negative integers to construct elements which we conjecture to lie in the annihilator ideal associated to the Galois action on the higher dimensional algebraic K -groups of the ring of integers in L . For abelian G our conjecture coincides with a conjecture of Snaith and thus generalizes also the well-known Coates–Sinnott conjecture. We show that our conjecture is implied by the appropriate special case of the equivariant Tamagawa number conjecture (ETNC) provided that the Quillen–Lichtenbaum conjecture holds. Moreover, we prove induction results for the ETNC in the case of Tate motives h 0 (Spec( L ))( r ), where r is a strictly negative integer. In particular, this implies the ETNC for the pair ( h 0 (Spec( L ))( r ), ), where L is totally real, r is a maximal order containing ℤ[ ] G , and will also provide some evidence for our conjecture.