Washington (CNN) As the nation mourns the death of John McCain, Arizona's Republican governor must soon decide who will replace him in the Senate.

Under Arizona law, it's up to Gov. Doug Ducey to fill Senate vacancies until a special election is held in 2020. He said in a statement that he would not name a successor until after McCain was buried later this week.

"Now is a time for remembering and honoring a consequential life well lived," Ducey said in the statement.

In the year since McCain was diagnosed with brain cancer, Ducey -- who faces a tough re-election race this year -- has refused to talk about who he might appoint to the seat.

Now required to fill the vacancy, Ducey faces major questions that could steer his decision. Among them: Will he seek a placeholder who won't run for re-election in 2020? And did McCain indicate who he'd like to fill the job?

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