The lifted root number conjecture for small sets of places

Let L / K be a finite Galois extension of number fields with Galois group G . The lifted root number conjecture (LRNC) by Gruenberg, Ritter and Weiss relates the leading terms at zero of Artin L ‐functions attached to L / K to natural arithmetic invariants. Burns used complexes arising from étale cohomology of the constant sheaf ℤ to define a canonical element T Ω( L / K ) of the relative K ‐group K 0 (ℤ G , ℝ). It was shown that the LRNC for L / K is equivalent to the vanishing of T Ω( L / K ) and that this, in turn, is equivalent to the equivariant Tamagawa number conjecture for the pair ( h 0 (Spec ( L ))(0), ℤ G ). These conjectures make use of a finite G ‐invariant set S of places of L that is supposed to be sufficiently large. We formulate an LRNC for small sets S that only need to contain the archimedean primes and give an application to a special class of CM‐extensions.