jurisdiction
This term carries a heavy weight of formality and legality, evoking an image of a boundary line—either geographic or thematic—that defines where a person's power ends and another's begins. It is rarely used in casual conversation unless referring to a specific legal dispute or a bureaucratic tug-of-war between agencies. While authority is a general quality of leadership, jurisdiction is a specific, granted right. It implies a rigid framework of laws and rules; if you are outside a jurisdiction, you are effectively invisible to that specific legal system, regardless of how much authority the official might possess personally.
Countable when referring to a specific geographic area governed by a court (the three jurisdictions of the tri-state area). Uncountable when referring to the abstract legal power to act (the judge lacks jurisdiction in this case).