WASHINGTON – The Supreme Court on Monday struck down significant portions of the controversial Arizona immigration law, but allowed to stand a hot-button provision that requires police officers to review the immigration status of any detainees they suspect of being in the United States illegally.

The high court, reviewing four portions of the Arizona law, struck down three of them, saying that the state had overstepped its authority by making it a state crime for immigrants to not register with the federal government, or for illegal immigrants to seek work or hold a job. It also threw out a provision in Arizona law that would allow police to arrest suspected illegal immigrants without a warrant.

Although the justices were divided, 5-3, on the three provisions, on the most controversial ruling they were united: They unanimously upheld a centerpiece of the legislation, the so-called “show me your papers” provision that requires police officers to check the immigration status of anyone stopped or arrested who they suspect is an illegal immigrant.

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Critics—including President Obama— say the provision will encourage racial profiling, because officers will check the status only of those who look like they are illegal immigrants.

The justices suggested the provision could be subject to more challenges, depending on how it is implemented.

Overall, the case tested a state’s powers to enforce federal immigration laws. The Supreme Court essentially ruled that it was up to Congress – not individual state houses—to solve some of the persistent problems the country faces around immigration.

“The national government has significant power to regulate immigration,” Justice Anthony Kennedy wrote for the five-member majority on the three provisions struck down. “With power comes responsibility, and the sound exercise of national power over immigration depends on the nation’s meeting its responsibility to base its laws on a political will informed by searching, thoughtful, rational civic discourse.”

“Arizona may have understandable frustrations with the problems caused by illegal immigration while that process continues, but the state may not pursue policies that undermine federal law,” he added.

The others voting in the majority were Chief Justice John Roberts, and Justices Stephen G. Breyer, Ruth Bader Ginsburg, and Sonia Sotomayor.

Justices Samuel A. Alito Jr., Antonin Scalia, and Clarence Thomas were the dissenting votes, while Justice Elena Kagan, who was solicitor general when the Obama administration sued Arizona, recused herself.

Scalia was particularly pointed in his dissent, at one point taking President Obama to task for a shift in policy earlier this month to halt deportations for certain young immigrants who came to the United States illegally with their parents.

“The president said at a news conference that the new program is ‘the right thing to do’ in light of Congress’s failure to pass the administration’s proposed revision of the immigration act,” Scalia wrote. “Perhaps it is, though Arizona may not think so. But to say, as the court does, that Arizona contradicts federal law by enforcing applications of federal immigration act that the president declines to enforce boggles the mind.”

The decision further injects the volatile topic of immigration into the presidential race, with both candidates trying to appeal to the Hispanic voters who are making up a larger slice of the electorate, particularly in key swing states such as Florida, Colorado, and Nevada, as well as Arizona.

Obama said he was “pleased” with most of the decision, but was concerned about the “show me your papers” ruling, agreeing with many immigration advocates that it would encourage racial profiling.

“No American should ever live under a cloud of suspicion just because of what they look like,” Obama said in a statement. “Going forward, we must ensure that Arizona law enforcement officials do not enforce this law in a manner that undermines the civil rights of Americans, as the Court’s decision recognizes.”

Romney, who took a hard-line stance on immigration during the Republican primary but has since softened his approach, lightly criticized the Supreme Court decision. During the Republican primary, Romney spoke in favor of the Arizona law.

“Given the failure of the immigration policy in this country, I would have preferred to see the Supreme Court give more latitude to states, not less,” Romney said at a prearranged fundraiser that was held in Scottsdale, Ariz.

Romney pledged that in his first year he would “make sure we actually do take on immigration.”

In Massachusetts, one of many states that erupted in controversy after Arizona passed its law two years ago, the Supreme Court decision generated praise, concern, and a measure of uncertainty.

After Arizona passed the law in 2010, the Boston City Council responded with a resolution divesting funds from the state, a largely symbolic move, and Governor Deval Patrick blasted the law and vowed to veto any similar legislation in Massachusetts.

But the Massachusetts Legislature has continued to debate budget amendments that would limit illegal immigration and last month Massachusetts, after protracted debate, fully deployed Secure Communities, a federal system that automatically checks the fingerprints of everyone arrested by state and local police against immigration databases.

Though more passive than having police question illegal immigrants, Secure Communities is allowing federal officials to screen the immigration status of tens of thousands of people arrested here and nationwide.

Observers on both sides of the debate said states are likely to pursue their own immigration laws as long as Congress and the president fail to address the issue.

“What’s happening is because Congress hasn’t created a clear and concise policy on this,” Bristol County Sheriff Thomas Hodgson said Monday. “This has gone on and on and on.”

The ruling also prompted responses from the US Senate candidates.

Senator Scott Brown, the Massachusetts Republican, said the decision “is another reminder that the federal government needs to deal with our broken immigration system.”

“I believe the first step is securing the border and turning off the magnets that encourage people to come into country illegally,” he said.

Elizabeth Warren, who is Brown’s Democratic rival, criticized Republicans generally and Brown specifically for not doing more on immigration.

Said Warren: “When he talks about ‘fixing the broken system,’ Scott Brown needs to acknowledge that he’s actively blocking reform.”