  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 !X/Y and also its
square root !1/2 X/Y . These formulas allow us us to carry out explicit
calculations; in particular, we are able to exhibit examples where these two
Euler characteristics and that of the 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].
