(CNN) Deputy Attorney General Rod Rosenstein told special counsel Robert Mueller in a classified August 2, 2017, memo that he should investigate allegations that President Donald Trump's former campaign chairman Paul Manafort was "colluding with Russian government officials" to interfere in the 2016 presidential election, prosecutors in the Russia probe revealed late Monday night.

Mueller was also empowered by Rosenstein to investigate Manafort's payments from Ukrainian politicians, a cornerstone of the Trump adviser's decades-long lobbying career that has resulted in several financial criminal charges so far.

The revelation of the August 2 memo comes amid a broader court filing from Mueller's prosecutors that offers a full-throated defense of their investigative powers and indictments thus far. In the filing, the special counsel's office argues that a federal judge should not throw out Manafort's case. Manafort has sought to have the case dismissed, arguing that the charges against him are outside of Mueller's authority.

The filing Monday night crystallizes the extent to which Rosenstein, who has come under fire by President Donald Trump and others, has backed the investigation's actions. (Rosenstein oversees Mueller's investigation following Attorney General Jeff Sessions' recusal.)

The memo, attached to Monday night's court argument and not previously disclosed even to Manafort, describes how Rosenstein's public order that appointed Mueller in May left out some details so it didn't confirm "specific investigations involving specific individuals."

Most of the investigations and individuals that Rosenstein named in that memo are now redacted -- amounting to almost a full page of withheld information.

But one section isn't redacted: The investigation Mueller could pursue against Manafort, with the Department of Justice's backing.

The special counsel's office was able to investigate two avenues of alleged crimes Manafort committed -- working with Russians to influence the election in violation of US law, and for crimes that arose from payments Manafort received from former Ukrainian politicians.

The Ukrainian connections and payments form the basis of the federal criminal charges Manafort currently faces.

The investigation into Manafort's alleged collusion with Russians has not resulted in publicly known charges.

Also in Monday night's filing, Mueller's prosecutors explain how they might try to connect Manafort's pre-campaign lobbying work to possible crimes during the 2016 campaign.

Mueller's investigatory mission "would naturally cover ties that a former Trump campaign manager had to Russian-associated political operatives, Russian-backed politicians, and Russian oligarchs. It would also naturally look into any interactions they may have had before and during the campaign to plumb motives and opportunities to coordinate and to expose possible channels for surreptitious communications. And prosecutors would naturally follow the money trail from Manafort's Ukrainian consulting activities."

Manafort has previously said the charges he faces in federal court in DC and Virginia have nothing to do with his work on the Trump campaign in 2016, and the White House has distanced Donald Trump's presidential campaign from Manafort. Manafort has pleaded not guilty in both cases.

Manafort had claimed the special counsel's office had no authority to bring the cases. "Each step the Special Counsel has taken against Mr. Manafort has been without lawful authority," Manafort's lawyers wrote in mid-March. The filing echoed an argument Manafort made in a related civil lawsuit against Mueller and the Justice Department, which also asks the court to throw out his charges. Arguments for that case are scheduled for this Wednesday, before the same judge overseeing the criminal case against Manafort.

But Mueller's team Monday night refutes Manafort's complaints in its 45-page legal brief. "Under this regime, a runaway Special Counsel is an impossibility," the prosecutors write, referring to current laws that regulate the office of a special counsel in the Justice Department.

Manafort is scheduled to face a jury in Virginia in July, then awaits trial in DC in September.