Duality and Hermitian Galois Module Structure

Abstract

Suppose $\mathcal{O}$ 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 $\mathcal{X}$ be a regular projective scheme which is flat and equidimensional over $\mathcal{O}$ of relative dimension $d$. Suppose $G$ is a finite group acting tamely on $\mathcal{X}$. Define ${\rm HCl}(\mathcal{O} G)$ to be the Hermitian class group of $\mathcal{O} G$. Using the duality pairings on the de Rham cohomology groups $H^*(X, \Omega^\bullet_{X / F})$ of the fiber $X$ of $\mathcal{X}$ over $F = {\rm Frac}(\mathcal{O})$, we define a canonical invariant $\chi_H(\mathcal{X}, G)$ in ${\rm HCl}(\mathcal{O} G)$ . When $d = 1$ and $\mathcal{O}$ is either $\mathbb{Z}$, $\mathbb{Z}_p$ or $\mathbb{R}$, we determine the image of $\chi_H(\mathcal{X}, G)$ in the adelic Hermitian classgroup ${\rm Ad\,HCl}(\mathbb{Z} G)$ by means of $\epsilon$-constants. We also show that in this case, the image in ${\rm Ad\,HCl}(\mathbb{Z} G)$ of a closely related Hermitian Euler characteristic $\chi_{H}(\mathcal{X}, G)(0)$ both determines and is determined by the $\epsilon_0$-constants of the symplectic representations of $G$.