P1684....Battery disconnected during last 50 starts
P0456....Evaporative Emission Control System Leak Detected (very small leak)
There is no one certain part to replace. The entire evaporative emissions systems has to be checked for leaks- it may be one of several parts, one of the multiple vacuum hoses, a faulty PCM, or a short/open in the leak detection pump.
The only way to check for a leak in the system is with a special smoke machine that inserts a dye smoke into the evap system under pressure and then checked for leaks. If none are found, then a factory scan tool has to be used to check each component with command functions to see if they respond