ε-constants and equivariant Arakelov Euler characteristics

Let R[G] be the group ring of a finite group G over a ring R. In this article, we study Euler characteristics of bounded metrised complexes of finitely generated Z[G]-modules, with applications to Arakelov theory and the determination of ǫ-constants. A metric on a bounded complex K• of finitely generated Z[G]-modules is specified by giving for each irreducible character φ of G a metric on the determinant of the φ-isotypic piece of the complex Q ⊗Z K • of Q[G]-modules. Ignoring metrics for the moment, the alternating sum of the terms of K• yields an Euler characteristic in the Grothendieck group G0(Z[G]) of finitely generated Z[G]-modules. If K • is perfect, in the sense that all its terms are projective, one has an Euler characteristic in the finer Grothendieck group K0(Z[G]) of all finitely generated projective Z[G]-modules. To take metrics into account, we will use a metrized version A(Z[G]) of the projective class group of Z[G]. We will construct in A(Z[G]) an “Arakelov-Euler characteristic” associated to each bounded perfect metrised complex of Z[G]-modules. Our primary interest will be metrised complexes arising in the following way. Let X be a scheme which is projective and flat over Spec(Z) and which has smooth generic fibre. We suppose that X supports an action by a finite group G and that the action is tame in the sense that, for each closed point x of X , the order of the inertia group of x is coprime to the residue characteristic of x. Choose a G-invariant Kähler metric h on the tangent bundle of the associated complex manifold X (C). We are then able to construct an ArakelovEuler characteristic for any hermitian G-bundle (F , j) on X by endowing the equivariant determinant of cohomology of RΓ(X ,F) with equivariant Quillen metrics jQ,φ for each irreducible character φ of G. This construction can be extended to give an Arakelov-Euler Supported in part by NSF grant DMS97-01411. Supported in part by NSF grant DMS99-70378 and by a Sloan Research Fellowship. EPSRC Senior Research Fellow.