Duality and Hermitian Galois Module Structure

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(O G ) to be the Hermitian class group of O G . 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 χ H (X, G ) in HCl(O G ). When d = 1 and O is either Z, Z p or R, we determine the image of χ H (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 χ H (X, G )(0) both determines and is determined by the ε 0 ‐constants of the symplectic representations of G . 2000 Mathematics Subject Classification 11G40, 11R33, 14G25