Euler-Poincare formula in equal characteristic under ordinariness assumptions

Let X be an irreducible smooth projective curve over an algebraically close d field of characteristic p > 0. Let F be either a finite field of characte ristic p or a local field of residue characteristic p. Let F be a construct ible etale sheaf of F-vector spaces on X. Suppose that there exists a finit e Galois covering pi: Y --> X such that the generic monodromy of pi*F is pr o-p and Y is ordinary. Under these assumptions we derive an explicit formul a for the Euler-Poincare characteristic X (X, F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck-Ogg-Sha fanvich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the cov ering pi, it is satisfied in interesting cases such as Drinfeld modular cur ves.