## 2006.353: Duality and Hermitian Galois Module Structure

2006.353:
Ted Chinburg, Georgios Pappas and Martin J. Taylor
(2003)
*Duality and Hermitian Galois Module Structure.*
Proceedings of the London Mathematical Society, 87 (1).
pp. 54-108.
ISSN 0024-6115

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DOI: 10.1112/S0024611502014016

## 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 $G$.

Item Type: | Article |
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Uncontrolled Keywords: | Galois structure; coherent cohomology; duality; Hermitian pairings. |

Subjects: | MSC 2000 > 11 Number theory MSC 2000 > 14 Algebraic geometry |

MIMS number: | 2006.353 |

Deposited By: | Miss Louise Stait |

Deposited On: | 21 August 2006 |

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