All of these voters appeared at the polling place for the precinct in which they were registered, and all of the signatures on their provisional-ballot envelopes matched the appropriate poll book signatures. At least 14 of these voters had voted in 10 elections before last year, according to voting records.

Last year, legislators in 27 states proposed laws like the one in Indiana, seeking to increase identification requirements for registration or voting, according to Justin Levitt, a lawyer for the Brennan Center for Justice at the New York University School of Law. Mr. Levitt predicted legislative and legal battles in at least that many states, depending on how the Supreme Court ruled.

Image Valerie Williams, a Republican, says she was similarly left out. Credit... AJ Mast for The New York Times

But the decision will affect more than just the voter-identification issue.

In the 1980s and ’90s, the Supreme Court came up with a test for assessing any law that placed hurdles before voters. The justices ruled that courts must weigh the value of the law to the state against the burden it placed on voters.

How the court applies that test in this case could set the standard for challenges to election rules across the country. The decision could affect a range of other voting-related rules being imposed by states, including ones involving the handling of provisional ballots, new restrictions on voter registration and the methods states can use to purge voters from registration rolls.

Professor Tokaji, who was hired by the federal Election Assistance Commission to help produce a report released last year on voter-identification requirements , said that Arizona, for example, would study the court’s ruling to see whether it could continue to require voters to present either one photo ID or two forms of nonphoto ID. Similarly, in Ohio, litigation is pending over such matters as how old a utility bill can be for a voter to use it as identification.

Supporters of the law said the requirement was hardly burdensome in today’s society.

“It is exceedingly difficult to maneuver in today’s America without a photo ID (try flying, or even entering a tall building such as the courthouse in which we sit, without one),” wrote Judge Richard A. Posner in a January 2007 opinion by the Seventh United States Circuit Court of Appeals in Chicago upholding the Indiana law and affirming a 2007 decision by Sarah Evans Barker, a federal district court judge. “And as a consequence, the vast majority of adults have such identification.”