On main conjectures in non-commutative Iwasawa theory and related conjectures
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 ℤp-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.
With the exception of the arithmetic results discussed in Section 2.3 and Section 9 this article is a long overdue update of a preprint first circulated in 2007. The general approach described here was presented in a seminar at the University of Kyoto in early 2006 and the author is extremely grateful to Kazuya Kato for his invitation to visit the University of Kyoto, for his interest and generous encouragement and for several very insightful remarks. He is also grateful to Jan Nekovář and Otmar Venjakob for many stimulating discussions and to the referee for numerous helpful comments.
Some of this article was written when the author held a Visiting Professorship at the University of Paris 6 in June 2007 and he is very grateful to the Institute of Mathematics for this wonderful opportunity. The results described in Section 2.3 and Section 9 were obtained just after Ritter and Weiss made [J. Amer. Math. Soc. 24 (2011), 1015–1050] available and shortly after we circulated (in April 2010) a version of this article including these arithmetic results Kakde obtained a more elegant proof of Theorem 9.1 (ii). (Kakde's proof is now given in [Invent. Math. 193 (2013), no. 3, 539–626] and uses our Theorem 2.1 as well as a more explicit variant of Theorem 6.1 and the general approach that we introduce in Remark 2.5 but does not rely on the main result of [J. Amer. Math. Soc. 24 (2011), 1015–1050].)
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Toric varieties, monoid schemes and cdh descent
- The kernel of the reciprocity map of varieties over local fields
- Tropical geometry over higher dimensional local fields
- Coleman–Gross height pairings and the p-adic sigma function
- On main conjectures in non-commutative Iwasawa theory and related conjectures
- Ahlfors–Weill extensions in several complex variables
- Double solid twistor spaces II: General case
- Symplectic geometry and rationally connected 4-folds
Articles in the same Issue
- Frontmatter
- Toric varieties, monoid schemes and cdh descent
- The kernel of the reciprocity map of varieties over local fields
- Tropical geometry over higher dimensional local fields
- Coleman–Gross height pairings and the p-adic sigma function
- On main conjectures in non-commutative Iwasawa theory and related conjectures
- Ahlfors–Weill extensions in several complex variables
- Double solid twistor spaces II: General case
- Symplectic geometry and rationally connected 4-folds
