A Local Approach to Chinburg's Root Number Conjecture

Suppose given a finite Galois extension K / k of number fields with group G and a finite sufficiently large G ‐stable set S of primes of K , write E for the group of S ‐units in K , and Δ S for the augmentation submodule of the Z‐free permutation module Z S on S . Let also A , B be finitely generated cohomologically trivial Z G ‐modules, and τ Tate's canonical class defining a 2‐extension E ↣ A → B ↠ Δ S . Chinburg has assigned the algebraic invariant Ω = [A]−[B] ∈ K 0 ZG to τ and conjectured that Ω equals the root number class W ( K / k ), an analytic invariant defined by Cassou‐Nogués and Fröhlich, and based on Artin's root numbers. We combine Ω with G ‐injections φ : Δ S ↣ E to obtain lifts Ω φ ∈ K 0 T ( Z G ) of Ω in the Grothendieck group of finite Z G ‐modules of finite projective dimension. Unlike K 0 Z G , the group K 0 T (Z G ) has a natural decomposition as a direct sum ⊕ K 0 T (Z l G ), over all finite rational primes l . This provides a local setting for the study of the Ω φ . Each of the above G ‐equivariant maps φ determines a function A φ on characters of G via L ‐values at zero. Stark's conjecture asserts that A φ is Galois‐equivariant. These functions can be viewed as naming elements of K 0T (Z G ) by using Fröhlich's Hom description. The new conjecture, referred to as the Lifted Root Number Conjecture, is that A φ names Ω φ . The truth of this conjecture would imply that of Chinburg's. The dependence of the invariant Ω φ on S , how it varies with φ, and its behaviour under restriction to subgroups and deflation to factor groups of G , are studied in individual sections. Then, the Hom description is used to compare Ω φ with A φ whenever Stark's conjecture holds true. It is shown that it suffices to give a proof of the Lifted Root Number Conjecture at the prime l in the case when the Galois group G is replaced by an l ‐elementary group, that is, the direct product of an l ‐group and a cyclic group of order prime to l . 1991 Mathematics Subject Classification : primary 11R42; secondary 11R33, 11R27.