Equivariant Euler characteristics and sheaf resolvents

Abstract

For certain tame abelian covers of arithmetic surfaces X/Y we obtain striking formulas, involving a quadratic form derived from intersection numbers, for the equivariant Euler characteristics of both the canonical sheaf ωX/Y and also its square root ω 1/2 X/Y . These formulas allow us to carry out explicit calculations; in particular, we are able to exhibit examples where these two Euler characteristics and that of the structure sheaf of X are all different and non-trivial. Our results are obtained by using resolvent techniques together with the local Riemann-Roch approach developed in [CPT]. Introduction Let N/K be a finite Galois extension of number fields with Galois group G. A number of interesting arithmetic modules may be associated to such an extension: the ring of algebraic integers ON of N ; the codifferent D−1 N/K of N/K; and, when the ramification subgroups of N/K are of odd order, the square root of the codifferent D N/K . Their structure as Galois modules has been studied extensively. When N/K is at most tamely ramified, they are all three locally free Z[G]-modules. It was proved in [T1] and [T2] that, for any tame Galois extension N/K, the classes of ON and D−1 N/K in Cl(Z[G]), the locally free class group of Z[G]-modules, are equal; that is to say, ON is a selfdual Z[G]-module. If in addition we suppose that G is of odd order, then in fact ON , D−1 N/K and D −1/2 N/K are all three free over Z[G]. This result for the ring of integers is a consequence of the Frohlich conjecture, which was proved in [T3]. The result for the square root of the codifferent was obtained in [ET]. In such a situation where one of the modules ON , D−1 N/K and D −1/2 N/K