Process for producing conjugated diene polymer, conjugated diene polymer, and rubber composition

A process for producing a conjugated diene polymer comprising a polymerization step wherein a conjugated diene polymer having a cis-1,4-bond content of 98.5 mass % or more and a ratio (Mw/Mn) of the weight average molecular weight (Mw) to the number average molecular weight (Mn) measured by gel permeation chromatography of 2.5 or less is obtained by carrying out a polymerization reaction in conjugated diene compounds using a catalyst composition containing the following components (a) to (c):

• component (a) a rare earth element-containing compound which comprises at least a rare earth element of an atomic number of 57 to 71 in the periodic table, or a reaction product of the rare earth element-containing compound with a Lewis base,
• component (b) an aluminoxane and an organoaluminum compound shown by a general formula AlR1R2R3, wherein R1 and R2 individually represent a same or different hydrocarbon group having 1 to 10 carbon atoms or a hydrogen atom, and R3 represents a hydrocarbon group having 1 to 10 carbon atoms which is the same as or different from R1 and R2, and
• component (c) an iodine-containing compound which comprises at least one iodine atom in the molecular structure.