On the \(\varepsilon\)-constants of arithmetic schemes (Q1392481)

Let \(X \rightarrow \text{Spec} (\mathbb{Z}_p)\) be a regular flat scheme over the \(p\)-adic integers. Assume that there is a tame action of a finite group \(G\) on \(X\). For any complex representation \(\rho\) we have a local constant \(\varepsilon(X, G, \rho)\) which enters into the functional equation for the corresponding \(L\)-function. Let \(Y = X/G\). The trace map \(\text{Tr}: {\mathcal O}_X \rightarrow {\mathcal O}_X\) defines a structure of Hermitian module on \({\mathcal O}_X\). It is possible to introduce a Pfaffian of this module (following a method by A. Fröhlich). The Pfaffian corresponds to a vertical divisor on \(Y\). This divisor depends also on a choice of a symplectic representation \(\rho\) of \(G\). Assume that \(\rho\) be of virtual dimension zero. Then the local constant is a rational number. The main theorem is an expression of the local constant in terms of intersection numbers between the Pfaffian and some relative classes on \(X\). These classes are defined as the Chern classes of the sheaf of differential forms on \(Y\) with logarithmic singularities along the closed fiber.