There is no evidence that Mr. Trump’s Facebook ads directly influenced the author of the manifesto, who wrote that his views “predate Trump” and posted the document on 8chan, an online forum known as a haven for extremists. But Mr. Trump, through his speeches, tweets and campaign ads, has elevated the idea of an “invasion,” once a fringe view often espoused by white nationalists, into the public discourse.

Some other Republican candidates have echoed Mr. Trump’s language in their own ads. “Let’s call this what it is — an invasion of our country,” read a recent Facebook ad for Tommy Tuberville, a former Auburn football coach who is running for Senate in Alabama. Other Republicans who have used the word “invasion” in Facebook ads include a candidate for governor in West Virginia and a candidate for Senate in North Carolina.

The cognitive linguist George Lakoff said the word “invasion” was a potent one for Mr. Trump to use because of what it allowed him to communicate. “If you’re invaded, you’re invaded by an enemy,” he said. “An invasion says that you can be taken over inside your own country and harmed, and that you can be ruled by people from the outside.”

[White extremist ideology is connected to some of the deadliest shootings worldwide in recent years.]

Mr. Lakoff added: “When he’s saying ‘invasion,’ he’s saying all of those things. But they’re unconscious. They’re automatic. They’re built into the word ‘invasion.’”

For the writer of the manifesto, the concept of an “invasion” had an additional, racist meaning: He promoted a conspiracy theory called “the great replacement,” which claims that an effort is underway to replace white people with nonwhite people.