(CNN) President Donald Trump used a false claim about former Vice President Joe Biden to explain to Ukrainian President Volodymyr Zelensky why Biden should be investigated.

Trump also made a confusing apparent reference to a conspiracy theory about CrowdStrike, the American cybersecurity firm that investigated the 2016 hack of the Democratic National Committee.

The claim about Biden

According to a White House transcript released Wednesday , Trump said to Zelensky during a phone call in July:

"There's a lot of talk about Biden's son, that Biden stopped the prosecution and a lot of people want to find out about that so whatever you can do with the Attorney General would be great. Biden went around bragging that he stopped the prosecution so if you can look into it...It sounds horrible to me."

Facts First: Joe Biden did not brag that he stopped any prosecution. Rather, he bragged that he had successfully pressured the Ukrainian government to fire a prosecutor who was widely criticized for failing to bring corruption prosecutions.

Trump's suggestion was that Biden had boasted of stopping the "prosecution" of the business dealings of Mykola Zlochevsky, the owner of a company, Burisma Holdings, for which his son Hunter Biden sat on the board of directors.

But there was no actual prosecution of Zlochevsky or Burisma -- just an investigation that had not produced any prosecutions and that does not seem to have been pursued vigorously.

There is no evidence Joe Biden ever tried to thwart this investigation, much less that he boasted about doing so.

What Biden said

Joe Biden, the administration's point man on the Ukraine file, pressured Ukrainian leaders to fire prosecutor general Viktor Shokin, who had faced international criticism for what was seen as an unwillingness to prosecute elite corruption.

Shokin, hired in 2015, was fired in 2016. Biden bragged about the saga at an event in January 2018 , recounting how he had threatened to withhold a billion-dollar US loan guarantee to Ukraine if Shokin was not removed.

At no point during Biden's story did he mention the investigation into Zlochevsky or Burisma. Further, there is no sign that Biden did anything to impede the investigation -- which may not even have been active at the time.

Vitaliy Kasko, a former deputy prosecutor who resigned from Shokin's office in February 2016, alleging that the office was itself corrupt, told Bloomberg this year that the investigation was dormant by the time of Joe Biden's intervention. Daria Kaleniuk, executive director of Ukraine's Anti-Corruption Action Center, told the Washington Post for a July article that "Shokin was not investigating."

In a 2015 speech, the US ambassador to Ukraine, Geoffrey Pyatt, blasted "corrupt actors within the Prosecutor General's Office" for working against anti-corruption efforts. One example Pyatt cited: the office's refusal to assist in a United Kingdom investigation into Zlochevsky

The claim about CrowdStrike

According to the document released by the White House, Trump also said to Zelensky: "I would like you to find out what happened with this whole situation with Ukraine, they say CrowdStrike...I guess you have one of your wealthy people...The server, they say Ukraine has it." (The ellipses are in the White House document.)

It's not entirely clear what Trump meant, but some inferences can be made based on his previous remarks.

CrowdStrike isn't Ukrainian, but Trump has said it is.

One of the company's senior executives previously had a lengthy career at the FBI . And one of its co-founders, Dmitri Alperovitch, is a senior fellow at the Atlantic Council, a prominent think tank in Washington. Alperovitch was born in the Soviet Union, in what is now Russia, then emigrated with his family to the US.

It isn't clear who Trump was referring to when he mentioned "one of your wealthy people," but in a 2017 interview with the Associated Press , he said he has heard CrowdStrike "is Ukrainian-based." It was not then and it is not now.

"I heard it's owned by a very rich Ukrainian, that's what I heard," Trump added in that 2017 interview, though, again, it is not and was not.

It is possible Trump was alluding to a very vague conspiracy theory involving wealthy Ukrainian businessman Viktor Pinchuk , a financial supporter of the Atlantic Council, where, again, Alperovitch is a senior fellow.

There is no evidence of any wrongdoing by Pinchuk.

CrowdStrike and the DNC

Trump did not specify to Zelensky what he meant by "the server," but he has repeatedly complained about how the Democratic National Committee and the FBI handled a hack into DNC servers in 2016.The DNC hired CrowdStrike in 2016 to investigate the hack.

After conducting an investigation, CrowdStrike publicly blamed the Russian government for the hack -- a finding later confirmed by special counsel Robert Mueller's investigation.

Trump and his Republican allies have fomented conspiracy theories about the hack since 2016. Trump has publicly rejected overwhelming evidence that Russia was the country responsible. And he has repeatedly questioned why the DNC hired CrowdStrike to investigate the breach instead of handing its servers over to the FBI for direct examination.

There is no evidence to support the conspiracies about the DNC servers. But by withholding the servers from the FBI, Democrats created a talking point for Republicans, who have claimed that the DNC was hiding something.

Former FBI director James Comey, who led the early phases of the Russia investigation, testified in 2017 that it would have been better if the DNC gave the servers to the FBI, but that the FBI was able to get the forensic information it needed from CrowdStrike. Mueller later brought detailed criminal charges against Russian hackers for their involvement in the hacks.

CrowdStrike said in a statement on Wednesday: "With regards to our investigation of the DNC hack in 2016, we provided all forensic evidence and analysis to the FBI. As we've stated before, we stand by our findings and conclusions that have been fully supported by the US Intelligence community."