With Gov. Jan Brewer sitting in the audience, the Supreme Court today heard the Obama administration’s challenge to Arizona’s strict immigration law with Chief Justice John Roberts suggesting at one point that the federal government “just doesn’t want to know who is here illegally.”

Solicitor General Donald B. Verrilli, who argued the health care case less than a month ago, had to field critical questions from the conservatives on the court, some of whom seemed eager to allow at least some provisions of the controversial state immigration law to go into effect.

Before Verrilli could launch into his argument, Chief Justice John Roberts said he wanted to “clear up at the outset” what the case in front of the justices was not about.

“No part of your argument has to do with racial or ethnic profiling, does it? ” Roberts asked Verrilli.

“That’s correct,” Verrilli answered. He reiterated that the issue before the court was whether the Arizona law — known as S.B. 1070 — interfered with existing federal law. Last year a lower court had ruled in favor of the Obama administration, blocking four of the most controversial provisions of the law from going into effect.

“Arizona is pursuing its own policy” of immigration control , Verrilli said. “It is our position the Constitution vests exclusive authority over immigration matters with the national government.”

But Justice Antonin Scalia, who seemed to be the most vocal supporter of the Arizona law, attacked. “All that means is that the government can set forth the rules concerning who belongs in this country. But if , in fact, somebody who does not belong in this country is in Arizona, Arizona has no power?” Scalia asked.

Roberts and Justice Samuel Alito zeroed in on one blocked provision of S.B. 1070 that requires Arizona law enforcement to ask someone they stop for their papers proving they are in the country legally if the officer has a reasonable suspicion that those individuals are in the country illegally. Law enforcement would later confer with the federal government on the legal status.

Roberts indicated that provision of the Arizona law does not interfere with federal law, but only notifies the federal government that someone is in the country illegally.

Verrilli responded that the state law forces the federal government to veer off from federal priorities . “Are we going to take our resources , which we deploy for removal” and divert them in a way that might be a “detriment to our priorities?” Verrilli asked.

At one point Roberts said, “It is not an effort to enforce federal law” because the decision to enforce the law is left up to the federal government. “It seems to me that the federal government just doesn’t want to know who is here illegally or not.”

Verrilli shot back, “No I don’t think that’s right. I think we want to be able to cooperate and focus on our priorities.”

Alito pointed out that while the Obama administration might choose to use its priorities to go after more dangerous offenders, the priorities of another administration might be different.

“You seem to be saying that what’s wrong with the Arizona law is that the Arizona legislature is trying to control what its employees are doing, and they have to be free to disregard the desires of the Arizona legislature for whom they work, and follow the priorities of the federal government, for whom they don’t work,” he said.

Verrilli tried again and again to reject the argument put forth by Arizona that SB 1070 worked cooperatively with existing federal law.

At one point Justice Sonia Sotomayor said about his argument, “You can see it’s not selling very well. Why don’t you try to come up with something else.”

The argument marked a rematch between Verrilli and Paul Clement, the lawyer who represented the 26 states challenging the health care law, and also represented Arizona in court today.

Clement said the government had taken the “extraordinary step” of challenging the state law even though it had been drafted to complement the federal law.

He was asked by several justices how certain provisions of the law would work and how the law enforcement provisions might affect those in the country legally, or even U.S. citizens.

Sotomayor asked about another one of the blocked provisions that allows warrantless arrests of a person if the officer has probable cause to believe the person to be arrested has committed an offense that makes the person deportable.

“Does section 6 permit an officer to arrest an individual who has overstayed a visitor’s visa by a day?” she asked.

Opponents of S.B. 1070 agreed that the conservatives on the court seemed to support the so called “show me your papers” provision of the law.

“But there appeared to be a majority of justices who had serious problems with other provisions of the law, including criminalizing failure to carry immigration papers and criminal sanctions for unauthorized migrants who seek work in Arizona,” says Elizabeth Wydra, chief counsel of the Constitutional Accountability Center.

After arguments, Gov. Brewer expressed satisfaction. Speaking to a throng of cameras she said, “I feel very confident as I walked out of there that we will get a favorable ruling in late June.”

Justice Elena Kagan did not participate in today’s arguments as she dealt with the issue in her previous job as solicitor general. The court is expected to decide this case around the same time it decides the health care case: near the end of June, just months before the next election.