At issue is the case of William Osborne, who was convicted in a brutal attack on a prostitute in Alaska 16 years ago. He won a federal appeals court ruling granting him access to a blue condom that was used during the attack. Testing its contents would firmly establish his innocence or guilt, says Osborne, who has admitted his guilt in a bid for parole.

But several justices said something more should be required, including a sworn declaration of innocence that would hold out the prospect of additional punishment for lying under oath.

One condition, Justice David Souter said, is that an "individual claiming the right to test claims that he is actually innocent."

People who waived DNA testing at the time of their trial also might not be able to test it later, some justices suggested.

The Obama administration, backing Alaska prosecutors, urged the court to reject the ruling of the federal appeals court in San Francisco in favor of Osborne.

Deputy Solicitor General Neal Katyal said the decision by the 9th U.S. Circuit Court of Appeals was too broad and would erase the requirement in the federal DNA testing law that a person must assert his innocence under penalty of perjury before getting access to the evidence.

Osborne has never made that claim and in fact admitted his guilt as part of his parole proceedings, Katyal said.

The New York state appeals panel issued a unanimous and unprecedented ruling in the case of Mark Speranza, 23, who left semen samples at the Repro Lab Inc. in July 1997 and signed a form directing that they be destroyed if he died. He wanted to be able to father a child if he survived his battle with cancer.

Following their son's death in January 1998, Mary and Antonio Speranza of Edison, N.J., told Repro's operator that they wanted a grandchild and wanted to save the sperm so a surrogate mother could be artificially inseminated.

The lab operator, Awilda Grillo, told the Speranzas their son deposited the specimens for his use only and the specimens had not been screened for donation to a member of the public, as required by state law.

The Speranzas paid Grillo storage fees and asked her to preserve the sperm specimens until a court could decide on custody.

State Supreme Court Justice Jane Solomon ruled that the law barred use of the sperm. She noted that the required screening, specifically a blood test of the donor, was now impossible since Mark Speranza was dead.

The emerald is one of the largest ever found and a judge conceded Tuesday it will take awhile to sort out all of the competing interests. The lawyer for one of the purported owners suggested outside court the gem may be a curse for his client. And the man who first reported it stolen said he was giving up his claims because "I have been threatened."

Larry Biegler, who refused to elaborate on the threats, told Los Angeles County Superior Court Judge John A. Kronstadt he was bowing out of the case. But it was his call to sheriff's deputies reporting that the emerald had been stolen that led to a multi-state caper in which two deputies trucked the mammoth object from Las Vegas.

Another of the claimants, Todd Armstrong of Eagle, Idaho, said outside court that he moved the emerald to Las Vegas after a gem dealer gave it to him as collateral for a shipment of diamonds he paid for but never received. He was trying to sell it when the law arrived.

The parties told the judge that the huge object is worth about $400 million. Lawyers said that museums including the Smithsonian and the Getty have expressed interest in obtaining it for their collections. Its value is as an art object and it can't be broken down to make jewelry, they said.

The emerald was dug up in Brazil in 2001 and became known as the Bahia Emerald. Its first owner was a Brazilian gem trader who made the first sale of it. But after that, its route became muddled. At one point, it wound up in a warehouse in New Orleans that was flooded during Hurricane Katrina.

The justices are expected to issue at least one ruling Tuesday morning, then consider a West Virginia case about when elected judges should step aside from cases that involve their campaign supporters.

At the heart of the dispute is $3 million that coal company executive Don Blankenship spent to help elect a state Supreme Court justice. Later, that justice — Brent Benjamin — sided with the coal company in a multimillion-dollar dispute with another mining firm.

The losing party, Harman Mining Corp., argues that Benjamin's refusal to step aside in the 3-2 decision violated the company's constitutional right to due process.

In an order Monday, the justices ended Marcus Borden's fight against the East Brunswick, N.J., school district's policy that forbids him and other staff members from joining in student-led prayer. The federal appeals court in Philadelphia sided with the district.

The high court declined to weigh in on whether Borden's desire to bow his head silently and "take a knee" with his football players violates the Constitution's prohibition on government endorsement of religion. Borden says such gestures are secular.

The school district says Borden, the East Brunswick coach since 1983, had a long history of leading prayers before he was ordered to stop after complaints from some parents. The district says the issue is whether its policy is constitutional, not Borden's actions.

Messages left for Borden and lawyer Ronald Riccio were not immediately returned Monday.

"Coaches are not supposed to be promoting religion; that's up to students and parents and pastors," said Barry W. Lynn, executive director of Americans United for Separation of Church and State, which represented the school district.

The 3rd U.S. Circuit Court of Appeals in Philadelphia agreed that the school district policy is constitutional, but the judges differed on what exactly the coach should do if his team prays.

The Supreme Court ended school-sponsored prayer in 1962 when it said directing that a prayer be said at the beginning of each school day was a violation of the First Amendment. The justices reaffirmed the decision in 2000 by saying a Texas school district was giving the impression of prayer sponsorship by letting students use loudspeakers under the direction of a faculty member for prayers before sports events.

The justices offer no comment on their action Monday, rejecting appeals in three separate cases, in favor of Dow Chemical, Monsanto and other companies that made Agent Orange and other herbicides used by the military in Vietnam.

Agent Orange has been linked to cancer, diabetes and birth defects among Vietnamese soldiers and civilians and American veterans.

The American plaintiffs blame their cancer on exposure to Agent Orange during the military service in Vietnam. The Vietnamese said the U.S.' sustained program to prevent the enemy from using vegetation for cover and sustenance caused miscarriages, birth defects, breast cancer, ovarian tumors, lung cancer, Hodgkin's disease and prostate tumors.

All three cases had been dismissed by the 2nd U.S. Circuit Court of Appeals in New York.

The appeals court said that lawsuit brought by the Vietnamese plaintiffs could not go forward because Agent Orange was used to protect U.S. troops against ambush and not as a weapon of war against human populations.

The other two suits were filed by U.S. veterans who got sick too late to claim a piece of the $180 million settlement with makers of the chemical in 1984. In 2006, the Supreme Court deadlocked 4-4 on whether those lawsuits could proceed.