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The lifted root number conjecture for small sets of places
Author(s) -
Nickel A.
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdp036
Subject(s) - mathematics , conjecture , invariant (physics) , equivariant map , sheaf , pure mathematics , galois group , galois module , combinatorics , discrete mathematics , mathematical physics
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.