An Arizona initiative several years ago took away the power to redistrict from the legislature and put it in the hands of an independent redistricting commission. The legislature is now challenging that this initiative as running afoul of the Times, Places and Manner Clause, which provides that the “manner of holding Elections for Senators and Representatives, shall be prescribed in each state by the legislature thereof.” (A relevant news story here.) The Arizona legislature’s argument is that such power, seized from the legislature by initiative, is prohibited.
One issue is that there are a handful of cases that have limited the power of the legislature: Davis v. Hildebrant (1916), which permitted a referendum to disapprove of legislative redistricting; and Smiley v. Holme (1932), which permitted a gubernatorial veto of legislative redistricting. But there is other precedent in other constitutional contexts in which the Court has construed “legislature” to mean not simply the law-making apparatus, but the legislature itself: McPherson v. Blacker (1892), which ensured that the legislature has plenary power over the manner of appointment of presidential electors; and Hawke v. Smith (1920), which held that a state constitution could not require that proposed constitutional amendments be put to a vote of the people, as that task was reserved to the legislature.
The Arizona legislature is arguing that the initiative is unlike a veto or referendum, because it essentially removes the legislature from the process altogether (except in some smaller ministerial ways, such as giving recommendations to the commission). But I think, on appeal, it would happily argue that the functional approach in Hildebrant and Smiley was incorrect, if it has the opportunity to do so on appeal.
The definition of "legislature" has been the subject of some scholarly debate, as Rick Hasen recently argued in the Hastings Constitutional Law Quarterly (PDF) that the definition of "legislature" may be the subject of some genuine debate in the context of the Electoral College and whether changes to the selection of electors could occur by initiative. Additionally, Franita Tolson has argued in the Utah Law Review that the legislature holding this power of redistricting helps states protect themselves in our federalist structure by giving them an additional mechanism to influence federal policy.
And courts have policed the word "legislature." In 2002, a three-judge panel in Smith v. Clark struck down a state court's congressional redistricting plan because the court lacked a "legislative" function under state law. (Thanks to Christopher Greene for flagging this case.)
So what about this case? The congressional election is fast approaching in Arizona, and a result in this case should be forthcoming soon. But the definition of "legislature" is not so simple as it might appear to be. Whether the Supreme Court will weigh in is, of course, another matter.