Toward equivariant Iwasawa theory

 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.