On the cohomologically trivial automorphisms of elliptic surfaces I: $\chi (S)=0$

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