Over the dissents of its three liberal members, the Supreme Court said on Tuesday that it would not hear an appeal from a juvenile offender in Arizona who was sentenced to life without the possibility of parole.

As is its custom, the court gave no reasons for turning down the case, which could affect dozens of prisoners.

The case concerns Lonnie Bassett, who was convicted of two murders committed in 2004, when he was 16. Mr. Bassett was riding in the back seat of a car when he shot the driver and a passenger.

When Mr. Bassett was sentenced in 2006, Arizona law did not give the judge the option of sentencing him to anything but life in prison without the possibility of parole.

But a 2012 Supreme Court decision, Miller v. Alabama, rejected mandatory sentences for youths who committed murders before they turned 18. To be constitutional, the court said, state laws must at least allow judges the option of sentencing juvenile offenders to life with the possibility of parole.

Still, the Arizona Supreme Court upheld Mr. Bassett’s sentence last year, saying that unusual features of Arizona’s sentencing law meant it did not fun afoul of the Miller decision.

In dissent, Justice Sonia Sotomayor, joined by Justices Elena Kagan and Ketanji Brown Jackson, said the state court’s decision violated the precedent.

“Arizona’s sentencing scheme left no discretion for a parole-eligible sentence in this case,” she wrote. “No one disputes that.”

The Arizona Supreme Court agreed but said that an idiosyncratic feature of the state law, allowing judges to choose between “natural life” without the possibility of release in any fashion and life without parole but with the theoretical possibility of clemency from the governor, rendered it constitutional.

Rejecting the usual understanding of the governing precedent, the court said, “Miller and its progeny do not specifically require the availability of parole when sentencing a juvenile offender.”

Lawyers for Mr. Bassett asked the U.S. Supreme Court to intervene, saying that his case could determine the fates of more than two dozen other juvenile offenders. The case, they wrote, “presents exceptionally significant questions about gamesmanship and the supremacy of federal law.”

In a brief urging the Supreme Court not to hear the case, Bassett v. Arizona, No. 23-830, Kristin K. Mayes, Arizona’s attorney general, chose not to defend the State Supreme Court’s reasoning.

“Parole eligibility is constitutionally required,” Ms. Mayes wrote. “This court’s precedents have already firmly established this fact, and Arizona does not disagree.”

She instead defended the result in the case on a different ground.

When Mr. Bassett was sentenced, she wrote, there was “a widespread mistaken belief among Arizona judges and attorneys” that one provision of the challenged state law allowed sentences of life with the possibility of parole. She added that “the mistaken belief appears to have been universal.”

Justice Sotomayor said that argument was misplaced.

“This argument requires speculating, based on no evidence, about the possibility of a judge’s two-decade-old mistaken belief about state law,” she wrote. “Miller permits no such thing.”