renig50, There could be a lot of "most likely" answers but you will need to do some testing unless you want to just start throwing money at it. Get a test light first. They are cheap and a great tool to have around a 52 year old wiring harness.
Now it very well could be the turn signal switch as Darter6 said. They are certainly famous for your type of issue but a turn signal switch is around $100 so a little testing first might help. Starting from the light the left turn signal/brake element is a dark green wire. The rear harness going forward has a connector which is under the left front kick panel. That connector connects to the dash harness. At least eliminate the rear light harness by looking for power at the connector, pin #3/dark green. From there it goes to the steering column/directional switch connector, pin#6, also dark green. If you don't get power at that point, get out your wallet.
Good luck and keep us posted.
http://www.mymopar.com/downloads/1965/65CoronetA.JPG
http://www.mymopar.com/downloads/1965/65CoronetB.JPG