Donald Trump signed a legal statement alleging voter fraud in the 2020 election despite being told the numbers underpinning the case were false, a federal judge said on Wednesday.

The disclosure was made by the US district judge David Carter, who ordered John Eastman, a former Trump lawyer, to provide more emails to the congressional committee investigating the January 6 attack on the US Capitol.

Eastman was one of Trump’s attorneys when the former president and his allies challenged his 2020 election loss to Joe Biden.

The legal document was part of a lawsuit by Trump’s team challenging the results in Georgia, a state Trump narrowly lost to Biden, in which they claimed a Georgia county had improperly counted more than 10,000 votes of dead people, felons and unregistered voters.

In an 18-page opinion, Carter said that the former president had “signed a verification swearing under oath” that the inaccurate fraud numbers were “true and correct” or “believed to be true and correct” to the best of his knowledge.

“The emails show that President Trump knew that the specific numbers of voter fraud were wrong but continued to tout those numbers, both in court and to the public,” the judge wrote, adding: “The Court finds that these emails are sufficiently related to and in furtherance of a conspiracy to defraud the United States.”

Carter previously ruled that Eastman and Trump had probably committed a felony by trying to pressure his then vice-president, Mike Pence, to obstruct Congress.

The ruling was made in a lawsuit filed by Eastman to block disclosure of the emails to the January 6 select committee, following a congressional subpoena.

Carter previously ordered Eastman to provide over 200 emails to the committee, after the lawyer resisted the subpoena and claimed that the communications were privileged.

The judge said on Wednesday that the vast majority of emails still being sought by congressional investigators should not be handed over, as legal protections given to attorneys and their clients apply to the records.

He said eight emails that would normally be shielded under those protections must be given to the committee, after he found that the communications were in furtherance of a crime – one of the few times those legal safeguards can be lifted.

Carter found that four emails show that Eastman and other lawyers suggested that the “primary goal” of filing lawsuits was to delay Congress’s certification of the 2020 election results.

The judge said four other emails “demonstrate an effort by President Trump and his attorneys to press false claims in federal court for the purpose of delaying the January 6 vote”.

At one point, Eastman wrote that although the former president had signed documents related to a lawsuit in Georgia on 1 December, “he has since been made aware that some of the allegations (and evidence proffered by the experts) has been inaccurate. For him to sign a new verification with that knowledge (and incorporation by reference) would not be accurate.”

Carter wrote: “President Trump and his attorneys ultimately filed the complaint with the same inaccurate numbers without rectifying, clarifying, or otherwise changing them.”

Trump and his allies filed over 60 lawsuits challenging the 2020 election, with some complaints alleging voter fraud without evidence to support those claims. Those cases were overwhelmingly rejected by judges, some of which Trump appointed to the federal courts.

The January 6 select committee last week voted to subpoena Trump in its investigation. It is expected to issue a report in the coming weeks on its findings.