THAINE'S THEOREM IN FUNCTION FIELD

Abstract

Let F be a finite real abelian extension of a global function field k with G = Gal(F/k). Assume that F is an extension field of the Hilbert class field Ke of k and is contained in a cyclotomic function field Kn. Let ' be any prime number not dividing phk|G|. In this paper, we show that if θ ∈ Z[G] annihilates the Sylow 'subgroup of O× F /CF, then (q − 1)θ annihilates the Sylow '-subgroup of ClF.

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