Process for polymerizing ethylene and copolymerizing ethylene with alpha-olefins and relevant catalyst

Ethylene homopolymers and ethylene copolymers with alpha-(C3-C10)-olefins are obtained by polymerizing the monomers with a polymerization catalyst formed by: a) an alkyl-aluminum; and b) a modified solid catalyst component obtained by treating with a Lewis' base a non-modified solid component to be defined by means of the following formula, expressed as atomic proportions: Ti (1), Hf (0.5-3), Si (1-6), Mg (0.5-15) Al (0,5-15), X (20-60), ROH (1-10) wherein: X represents either chlorine and bromine, and R represents an either linear or branched alkyl radical of from 1 to 6 carbon atoms. The catalyst is particularly useful for controlling the molecular weight and the molecular weight distribution of ethylene homopolymers and copolymers.