On the “main conjecture” of equivariant Iwasawa theory

By J. Ritter, Alfred Weiss. 2010

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We prove the `main conjecture' of equivariant Iwasawa theory when μ = 0 . [ A word of caution : Theorems 1, 2, 3, 4, 5 here are Theorems 1, 2, 5, 3, 4, respectively, in JAMS . ] The `main conjecture' of equivariant Iwasawa theory concerns the situation where l is a xed odd prime number , K/k is a Galois extension of totally real number elds with k/Q and K/k∞ nite, where k∞/k is the cyclotomic Zl-extension (we set G = G(K/k) and Γk = G(k∞/k)) , S a xed nite set (which will normally be suppressed in the notation) of primes of k containing all primes which ramify in K and all archimedean primes , and M is the maximal abelian l-extension of K unrami ed outside S (we set X = G(M/K)) . It asserts that a canonical re nement f = fS of the Iwasawa module X is determined by the Iwasawa L-function LK/k = LK/k,S of K/k. The data f and LK/k (and Hom∗(Rl(G), (QcΓk)×) below) have been de ned in [RW2, p.562/563 1, p.558]; we will not repeat the de nitions now (only in Ÿ1) but brie y explain how LK/k should determine f. Denote by ΛG , QG the completed group ring Zl[[G]] of G over Zl and its total ring of fractions, respectively. The localization sequence of K-theory → K1(ΛG) → K1(QG) ∂ →K0T (ΛG) → K0(QG) → has f in K0T (ΛG). The reduced norms of the Wedderburn components of the semi-simple algebra QG induce the map Det : K1(QG) → Hom∗(Rl(G), (QcΓk)×) . The Iwasawa L-function LK/k of K/k is derived from the S-truncated l-adic Artin L-functions and belongs to the above group Hom∗. The equivariant `main conjecture' asserts that there is a unique element Θ ∈ K1(QG) satisfying Det(Θ) = LK/k and, moreover, that this Θ has ∂(Θ) = f . In other words, Θ is the nonabelian generalization to K/k of the pseudomeasure of an abelian eld extension. ∗We acknowledge nancial support provided by DFG and NSERC. 1where f is denoted f

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/98172df9150c227f0e6729c526193de58047ca2d

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