The BATFE has historically used the grip frame serial number as the registered part of a Luger pistol. Probably because MOST automatic pistols have their full serial number recorded on that part.
The carbine is a special case as Lugerdoc has described.
In my own humble opinion, the BATFE has always been wrong on this point because they have always felt that the part of the handgun that controls or allows the weapon to be fired is the grip
FRAME. (Corrected based on Ron Wood's fabulous powers of discernment and his posting below, and my obvious error
)
The Luger pistol is unique to the best of my knowledge in that a complete upper receiver is all that is required to fire a shot... merely by loading a cartridge in the chamber, closing the toggle and depressing the sear bar manually...
WARNING ! ! ! ! ! ! ! ! DO NOT TRY THIS AT HOME... Don't ask how I know 
...even in a padded gloved hand, this would not be a comfortable method for discharging a 9mm round.
Yet the BATF allows the top half of a Luger pistol to move in interstate commerce between unlicensed persons... but requires the frame to be transferred only between licensed FFL and C&R holders... but of course that makes as much sense as much of what the federal bureaucracy does
