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.