The Supreme Court has interpreted the AUMF with the assumption that Congress intended for the President to pursue the conflict in accordance with traditional law-of-war principles, and has upheld the detention of a “narrow category” of persons who fit the traditional definition of “enemy combatant” under the law of war.
Other courts have been willing to accept a broader definition of “enemy combatant” to permit the detention of individuals who were not captured in circumstances suggesting their direct participation in hostilities against the United States, but a plurality of the Supreme Court warned that a novel interpretation of the scope of the law of war might cause their understanding of permissible executive action to unravel.
Consequently, Congress may be called upon to consider legislation to support the full range of authority asserted by the executive branch in connection with the GWOT [Global War on Terror]. In the event the Court finds that the detentions in question are fully supported by statutory authorization, whether on the basis of existing law or new enactments, the key issue is likely to be whether the detentions comport with due process of law under the Constitution.
The report notes that:
The [Supreme] Court did not foreclose all options available to Congress to streamline habeas proceedings involving detainees at Guantanamo or elsewhere in connection with terrorism.
It then catalogs the pending bills regarding "extra-territorial" rendition and detention. One gives the power to order rendition and detention to the FISA court.
Certain types of renditions and detentions appear to be excluded from these general requirements, including those of persons detained by the United States in Guantanamo on the act’s date of enactment who are transferred to a foreign legal jurisdiction, as well as the rendition or detention of individuals detained or transferred by the U.S. Armed Forces under circumstances governed by, and in accordance with, the Geneva Conventions.
Otherwise, extraterritorial detention would require the authorization of the President or the Director of National Intelligence based on a certification that the failure to detain that individual “will result in a risk of imminent death or imminent serious bodily injury to any individual or imminent damage to or destruction of any United States facility” or the factual basis exists to demonstrate the individual is an international terrorist and there is reason to believe that the detention or rendition of such person is important to the national security of the United States. An application for detention must be submitted to the FISC within 72 hours in order to detain the person.
Shorter version: The President can't abridge habeas corpus on his own, but if Congress agrees, so be it?