A LOCAL APPROACH TO CHINBURG'S ROOT NUMBER CONJECTURE

Abstract

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 $\Delta S$ for the augmentation submodule of the $\Bbb{Z}$-free permutation module $\Bbb{Z} S$ on $S$. Let also $A$, $B$ be finitely generated cohomologically trivial $\Bbb{Z} G$-modules, and $\tau$ Tate's canonical class defining a 2-extension $E\rightarrowtail A\to B\twoheadrightarrow\Delta S$. Chinburg has assigned the algebraic invariant $\Omega=[A]-[B]\in K_0(\Bbb{Z}G)$ to $\tau$ and conjectured that $\Omega$ 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 $\Omega$ with $G$-injections $\varphi:\Delta S\rightarrowtail E$ to obtain lifts $\Omega_\varphi\in K_0T(\Bbb{Z}G)$ of $\Omega$ in the Grothendieck group of finite $\Bbb{Z}G$-modules of finite projective dimension. Unlike $K_0(\Bbb{Z}G)$, the group $K_0T(\Bbb{Z}G)$ has a natural decomposition as a direct sum $\bigoplus K_0T(\Bbb{Z}_lG)$, over all finite rational primes $l$. This provides a local setting for the study of the $\Omega_\varphi$.

Each of the above $G$-equivariant maps $\varphi$ determines a function $A_\varphi$ on characters of $G$ via $L$-values at zero. Stark's conjecture asserts that $A_\varphi$ is Galois-equivariant. These functions can be viewed as naming elements of $K_0T(\Bbb{Z}G)$ by using Fröhlich's Hom description. The new conjecture, referred to as the Lifted Root Number Conjecture, is that $A_\varphi$ names $\Omega_\varphi$. The truth of this conjecture would imply that of Chinburg's.

The dependence of the invariant $\Omega_\varphi$ on $S$, how it varies with $\varphi$, 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 $\Omega_\varphi$ with $A_\varphi$ 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 $l$.