Toward equivariant Iwasawa theory

Abstract.

 Let l be an odd prime number, K/k a finite Galois extension of totally real number fields, and G _∞ , X _∞ the Galois groups of K _∞ /k and M _∞ /K _∞ , respectively, where K _∞ is the cyclotomic l-extension of K and M _∞ the maximal abelian S-ramified l-extension of K _∞ with S a sufficiently large finite set of primes of k. We introduce a new K-theoretic variant of the Iwasawa ℤ[[G _∞ ]]-module X _∞ and, for K/k abelian, formulate a conjecture, which is the main conjecture of classical Iwasawa theory when lł[K : k]. We prove this new conjecture when Iwasawa's μ-invariant vanishes and discuss consequences for the Lifted Root Number Conjecture at l.