The theory of \(\varepsilon\)-constants can be traced back to Gauss: the sign for the quadratic Gauss sums has important consequences for the distribution of quadratic residues, and Gauss sums occur as constants in the functional equations of Dirichlet \(L\)-series. To describe modern conjectures descending from these results, suppose \({\mathcal X}\) is a projective and flat regular scheme over Spec\(({\mathbb Z})\) which is an integral model of a smooth projective variety \(X\) of dimension \(d\) over \({\mathbb Q}.\) The Hasse-Weil function \(L({\mathcal X}, s)\) is conjectured to have an analytic continuation and to satisfy a functional equation \(L({\mathcal X}, s) = \varepsilon ({\mathcal X})\;A({\mathcal X})^{-s}\;L({\mathcal X}, d+1-s),\) where the “\(\varepsilon\)-constant” \(\varepsilon ({\mathcal X})\) and the “conductor” \(A({\mathcal X})\) are real numbers which, assuming certain choices, can be defined independently of any conjecture. The knowledge of these numbers is important in many arithmetic applications. Deligne’s formulae give expressions for both of them as products of certain rational numbers, roots of unity and generalized Gauss sums. Bloch’s conjectural conductor formula gives \(A({\mathcal X})\) as the degree of a localized Chern class.
In this paper, the authors study a “tame equivariant” situation: let \({\mathcal X} \to {\mathcal Y}\) be a tame \(G\)-cover of regular arithmetic varieties over \({\mathbb Z},\) \(G\) being a finite group; assuming that \({\mathcal X}\) and \({\mathcal Y}\) have “tame” reduction, they show how to determine the \(\varepsilon\)-constant in the conjectural functional equation of the Artin-Hasse-Weil function \(L({\mathcal X}/{\mathcal Y}, V, s)\) for a symplectic representation \(V\) of \(G\) from a suitably refined equivariant Arakelov-de Rham-Euler characteristic of \({\mathcal X}.\) Their result may be viewed firstly as a higher dimensional version of the Cassou-Noguès-Taylor characterization of tame symplectic Artin root numbers in terms of rings of integers with their trace form, and secondly as a signed equivariant version of Bloch’s conductor formula.