No Ignition

For a couple of years had an occasional problem where there was no ignition at all when the car was warm, usually from the SUN at first. After a minute or two it would suddenly go back to normal! Recently this became more frequent when the car had been driven and was still warm when starting, and then permenant! I had a mechanic look at it (fuel pump ok, starter ok, tried switching off immobiliser from door lock!?) to no avail. An AA guy then connected it to all his computer gadgetry but also has failed to find cause of the the problem. There is no power to Starter or Plugs! What could it be please??