The Greek noun hēgemonia (ἡγεμονία) means rule, leadership, sovereignty, authority, governorship. The abstract noun from hēgemōn (leader, governor), it appears once in the NT — Luke 3:1 — where it is translated "reign" (ESV: "in the fifteenth year of the reign of Tiberius Caesar"). The word describes the sphere and exercise of governing authority.
Luke 3:1 is remarkable in its historical density: the reign (hēgemonia) of Tiberius Caesar, the governorship of Pilate, the tetrarchies of Herod and Philip and Lysanias, and the high priesthood of Annas and Caiaphas — six layers of human authority — are the backdrop against which "the word of God came to John the son of Zechariah in the wilderness." God bypasses all the centers of political and religious hēgemonia and speaks to a man in the desert. The greatest governmental authority of the ancient world — Rome — and the most powerful religious institution — the Jerusalem priesthood — are mentioned only as temporal markers. The divine word moves where it will, often outside the corridors of established power.