"I mean, what was he doing with Lee Harvey Oswald shortly before the death, before the shooting? It's horrible," Trump said, referring to a photograph of Oswald with another man who a tabloid claimed was Cruz's father.

As is often his style, Trump did not make an explicit accusation or a precise charge, instead suggesting obliquely that someone in power is connected to a public tragedy and that law enforcement and the press are deceiving the public about what really happened.

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Many of Trump's supporters are more likely to believe this kind of rhetoric, according to a poll out Wednesday that showed that large numbers of Trump's supporters believe in conspiracy theories.

Although he had flirted with politics before, Trump's political career began in earnest in 2011, when he became the unofficial spokesman for the movement to discredit President Obama's U.S. birth. He still occasionally implies that Obama is not putting American interests first and that he might be an agent of some foreign, hostile power.

For example, Trump said in South Bend, Ind., on Monday that when it comes to combating violent Islamists, the president doesn't know what he's doing — "or maybe he does know what he's doing," Trump added.

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A large majority of Trump's coalition is suspicious of Obama's origins. Among respondents who chose Trump over Democratic front-runner Hillary Clinton, 40 percent said it was "definitely true" that "President Obama is hiding important information about his background and early life," according to Wednesday's poll, conducted at Fairleigh Dickinson University in New Jersey. Another 37 percent said the claim was possibly true.

When pollsters asked about Clinton and Cruz, 32 percent of respondents who chose Cruz said this claim about Obama was definitely true. Among Ohio Gov. John Kasich's supporters in a hypothetical contest with Clinton, that figure was just 28 percent.

Trump's supporters are also more likely to say that vaccines cause autism, a claim he has made a few times during the campaign. Fifty-two percent of his supporters said this claim was possibly or definitely true, compared to 49 percent of those who supported Cruz and 45 percent of those who supported Kasich in hypothetical contests between each Republican and Clinton.

(Despite extensive studies, there is no scientific evidence that vaccines cause negative health effects. On the other hand, inoculations are estimated to have saved the lives of 732,000 kids over the past two decades.)

Trump's supporters are even more likely to believe that the massacre at an elementary school in Newtown, Conn., was staged to motivate public support for gun control. It is a theory so implausible, not to mention offensive to the families of the victims, that Trump actually disavowed it through a spokeswoman.

Although Trump hasn't given credence to this view, 21 percent of his supporters told Fairleigh Dickinson said that it was possibly or definitely true. The figure was 20 percent among Cruz's supporters and 17 percent among Kasich's supporters.

That finding suggests Trump's endorsement, implicit or explicit, of other outlandish theories makes him a popular candidate among those who tend to be extremely mistrustful in general, said political science experts.

Belief in conspiracy theories is not a partisan trait. A majority of Democrats, for example, told Fairleigh Dickinson that it was at least possible that President George W. Bush knew about the terrorist attacks of Sept. 11, 2001 before they occurred.

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But previous polls by The Washington Post and ABC News confirm that distrust of government, broadly speaking, is one of the traits that most clearly separates Trump's supporters from other Republicans. Moreover, Fairleigh Dickinson's polls have consistently found that trust in the government is closely associated with the propensity to believe in many conspiracy theories, said Dan Cassino, a political scientist there.

"If you don’t trust the government, you’re more likely to believe they’re up to something nefarious," he added. Conspiracy theories are part of a broader worldview for those who suspect that everyone who is not an outright liar could be deluded or intimidated into shutting their eyes to the truth.

"Trump’s issue positions that have shifted incredibly over time, but the one thing that’s remained absolutely consistent is his penchant for conspiracy theorizing. Instead of using policy issues to try and garner supporters, he has used people’s fears and cynicism," said Joseph Uscinski, a political scientist at the University of Miami who was not involved in the poll. "He’s appealing to people who have very dark opinions."

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The poll also found that Trump's supporters were more likely to believe that global warming is a hoax fabricated by the scientific community and that Clinton knew that the U.S. diplomatic mission in Benghazi was going to be attacked and did nothing to prevent the killings there in 2012.

Among Trump's supporters in a hypothetical contest against Clinton, 29 percent said global warming was definitely a fabrication. This figure was 25 percent among Cruz's supporters and 22 percent among Kasich's supporters.

Likewise, 50 percent of Trump's supporters said Clinton definitely knew about the attack in Benghazi beforehand. Just 43 percent of Cruz's supporters and 40 percent of Kasich's supporters agreed. Trump is by no means the only Republican to have suggested that there might be some kind of conspiracy in connection with Benghazi or with global warming, but he does the best among Republicans who share those views.

On Wednesday, Trump denied that he ever thought that Cruz's father might have helped kill an American president. He also said, though, that others should decide for themselves.

"Of course I don’t believe that," he told CNN. "I don’t believe it, but I did say let people read it."