Soon after takeoff the crew were alerted to a pitch computer fault in the aircraft. As other indications were normal the pilot elected to investigate the problem when the aircraft was further away from the terminal area. The fault indication was reset. A short time later the same fault re-appeared in relation to another pitch computer, together with all other associated fault indications. The crew also became aware that the aircraft control mode had changed to alternate law. As weather conditions in the departure area were good they decided to return for a landing at the departure aerodrome. The pitch computer involved in the first fault indication was removed and tested. No fault was found. After further checking, the pitch trim control actuator (the component common to both pitch computers in normal operations) was replaced. No further faults were found and the aircraft was returned to service.