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Non-abelian pseudomeasures and congruences between abelian Iwasawa L-functions

By Jürgen Ritter and Alfred Weiss

Abstract

Abstract. The paper starts out from pseudomeasures (in the sense of Serre) which hold the arithmetic properties of the abelian l-adic Artin L-functions over totally real number fields. In order to generalize to non-abelian l-adic L-functions, these abelian pseudomeasures must satisfy congruences which are introduced but not yet known to be true. The relation to the “equivariant main conjecture ” of Iwasawa theory is discussed. Fix an odd prime number l and a finite field extension k/Q with k totally real. Let k ∞ be the cyclotomic Zl-extension of k and K ⊃ k ∞ be a totally real Galois extension of k with Galois group G and so that [K: k∞] is finite. Setting H = GK/k ∞ the group extension 1 → H → G → Γk → 1. and Γk = G k∞/k, we get Let first G be abelian. We consider the group algebra QG = Quot(ΛΓ)[H] which results from a splitting of the above group extension, with ΛΓ ( ≃ Zl[[T]]) the Iwasawa algebra of a preimage Γ of Γk in G. From Serre’s interpretation [Se] of the work [DR] of Deligne and Ribet on abelian L-functions over totally real fields it follows that there is a unique element λ K/k ∈ QG that encodes all the l-adic L-functions Ll(s,χ) of k∞/k for the characters χ of

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