# Duality and Hermitian Galois Module Structure

Chinburg, Ted and Pappas, Georgios and Taylor, Martin J. (2003) Duality and Hermitian Galois Module Structure. Proceedings of the London Mathematical Society, 87 (1). pp. 54-108. ISSN 0024-6115

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 $G$.