On the Cohomologically Trivial Automorphisms of Elliptic Surfaces I: $\chi(S) = 0$

In this first part we describe the group Aut(Z)(S) of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface S with Kodaira dimension kappa(S) = 1), in the initial case chi(O-S) = 0. In particular, in the case where Aut(Z)(S) is finite, we give the upper bound 4 for its cardinality, showing more precisely that if Aut(Z)(S) is nontrivial, it is one of the following groups: Z/2, Z/3, (Z/2)(2). We also show with easy examples that the groups Z/2, Z/3 do effectively occur. Respectively, in the case where Aut(Z)(S) is infinite, we give the sharp upper bound 2 for the number of its connected components.