Final answer:
The Supreme Court allowed the 'show me your papers' provision of Arizona's SB 1070 to stand, which lets law enforcement check immigration status during lawful stops if there's reasonable suspicion of undocumented status.
Step-by-step explanation:
The aspect of Arizona's 2010 immigration law, often called SB 1070, that the Supreme Court allowed to stand was the "show me your papers" provision. This provision authorizes police officers to check the immigration status of anyone they stop, detain, or arrest if they have reasonable suspicion to believe the person is an undocumented immigrant. While the Supreme Court struck down other provisions, including making it a crime for undocumented immigrants to seek work and to be in Arizona without valid immigration papers, it upheld the requirement for law enforcement to verify immigration status under certain conditions, though they warned against racial profiling.