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

Abstract

Abstract The main conjecture of non-commutative Iwasawa theory is shown to be equivalent, modulo an appropriate torsion hypothesis, to a family of main conjectures over cyclotomic ℤ p -extensions together with suitable compatibility relations between the associated abelian p-adic L-functions. This result combines with techniques of Ritter and Weiss and of Kakde to give a concrete strategy for deducing main conjectures for (non-abelian) compact p-adic Lie extensions of arbitrarily large rank from main conjectures in the classical sense. As a first application of this approach we combine it with a recent result of Ritter and Weiss to prove, modulo an appropriate vanishing hypothesis on μ-invariants, the main conjecture of non-commutative Iwasawa theory for Tate motives over compact p-adic Lie extensions of totally real number fields of arbitrarily large rank. We then also deduce several interesting consequences of this result concerning a range of related conjectures including special cases of the equivariant Tamagawa number conjecture.