Duality and Hermitian Galois Module Structure

Abstract

Suppose O is either the ring of integers of a number field, the ring of integers of a p-adic local field, or a field of characteristic 0. Let X be a regular projective scheme which is flat and equidimensional over O of relative dimension d. Suppose G is a finite group acting tamely on X. Define HCl(OG) to be the Hermitian class group of OG. Using the duality pairings on the de Rham cohomology groups H ∗ ( X , Ω X / F ∙ ) of the fiber X of X over F = Frac(O), we define a canonical invariant χH(X, G) in HCl(O G). When d = 1 and O is either Z, Zp or R, we determine the image of χH(X, G) in the adelic Hermitian classgroup Ad HCl}(Z G) by means of ε-constants. We also show that in this case, the image in Ad HCl(Z G) of a closely related Hermitian Euler characteristic χH(X, G)(0) both determines and is determined by the ε0-constants of the symplectic representations of G. 2000 Mathematics Subject Classification 11G40, 11R33, 14G25