Prime Number Conjecture

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 ll[K : k]. We prove this new conjecture when Iwasawa's μ-invariant vanishes and discuss consequences for the Lifted Root Number Conjecture at l.