# $\varepsilon$-constants and equivariant Arakelov-Euler characteristics

Chinburg, Ted and Pappas, Georgios and Taylor, Martin J. (2002) $\varepsilon$-constants and equivariant Arakelov-Euler characteristics. Annales Scientifiques De L'Ecole Normales Superieure, 35 (3). pp. 307-352. ISSN 0012-9593

Let $\mathcal{X} \rightarrow \mathcal{Y}$ be a tame $G$-cover of regular arithmetic varieties over $\Z$ with $G$ a finite group. Assuming that $\mathcal{X}$ and $\mathcal{Y}$ have “tame” reduction we 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 $V$ a symplectic representation of $G$ from a suitably refined equivariant Arakelov–de Rham–Euler characteristic of $\mathcal{X}$. Our result may be viewed firstly as a higher dimensional version of the Cassou-Noguès–Taylor characterization of tame symplectic Artin root numbers in term of rings of integers with their trace form, and secondly as a signed equivariant version of Bloch's conductor formula.