WASHINGTON — With President Donald Trump’s impeachment trial getting underway, Democrats are intensifying their demands for more testimony and documents that could add to the already voluminous evidence against him and bolster their case by shedding new light on several key questions.

Despite the White House strategy of blocking testimony from top officials and rejecting demands for documents, the Senate will have in front of it various accounts of how Trump eagerly sought to persuade Ukraine’s new president to pursue investigations into two matters that could benefit him in his reelection campaign. Those matters are dealings in Ukraine involving former Vice President Joe Biden and his son Hunter Biden, and purported Ukrainian meddling in the 2016 U.S. presidential election.

But in part because of the White House’s decision not to cooperate, the record of actions by Trump and his underlings is riddled with gaps — and new evidence has been surfacing at the eleventh hour. Testimony from Trump’s senior advisers could illuminate how overt the president’s efforts were, though it is unclear if that would persuade any Republican senators to abandon their defense of the president.

On Sunday, Rep. Adam Schiff, chairman of the House Intelligence Committee and the lead House impeachment manager, said he was concerned that the CIA and National Security Agency were withholding information about Ukraine out of fear of angering the president.

“The NSA in particular is withholding what are potentially relevant documents to our oversight responsibilities on Ukraine but also withholding documents potentially relevant that the senators might want to see during the trial,” Schiff, D-Calif., said on ABC’s “This Week,” referring to the National Security Agency.

Republicans called those complaints proof that the case against Trump was so weak that Democrats were scrambling to bolster it. “But this, to me, seems to undermine or indicate that they’re getting cold feet or have a lack of confidence in what they’ve done so far,” Sen. John Cornyn, R-Texas, said Sunday on “Face the Nation.”

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Even before new information emerged in recent days — including material from Lev Parnas, who worked closely with the president’s personal lawyer, Rudy Giuliani, to seek damaging information about the Bidens and to impugn the U.S. ambassador in Kyiv, Ukraine — the Senate’s 100 jurors faced unanswered questions that go to the heart of the matter.

What exactly did John Bolton, then the White House’s national security adviser, see and hear last year that convinced him a group of diplomats and aides were cooking up a geopolitical “drug deal” involving Ukraine?

How often and how thoroughly did Giuliani, the chief engineer of the pressure campaign, brief the president on what he was up to? What has Trump said behind closed doors about his order to freeze military aid to Ukraine?

Democrats in the Senate want to call Bolton to the stand; compel the testimony of three other top Trump aides, including the acting chief of staff, Mick Mulvaney; and obtain the records that the administration has withheld. But they would need the support of at least four Republican senators to do so.

“A fair trial, everyone understands, involves evidence,” Sen. Dick Durbin, D-Ill., said Sunday on NBC’s “Meet the Press.” “Evidence would be documents and witnesses. We know the president has refused to provide documentation beyond the July 25 telephone memo. And he’s refused to provide basic witnesses who actually heard what happened on that conversation and saw what happened afterwards.”

Impeachment and President Trump

Here are some of the key questions that more witness testimony or additional documents could address.

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To what extent did Giuliani act at Trump’s direction and keep him apprised of his efforts?

Some Trump allies have tried to suggest that Giuliani was a rogue actor pursuing his own interests in Ukraine. But Giuliani said last spring, as he planned a trip to Ukraine to press for investigations into the Bidens, that his efforts had Trump’s full support and that the president “basically knows what I’m doing.”

Trump has called Giuliani a “crime fighter” who was “seeking out corruption” because he was “very, very incensed at the horrible things that he saw.” He also said Giuliani had the right to look into whether the Ukrainians helped sow the seeds for the special counsel’s investigation into Russia’s interference in the 2016 election.

Giuliani has insisted that his conversations with Trump are protected by attorney-client privilege. He has pointed out that Kurt Volker, then Trump’s special envoy to Ukraine, put him in touch with a top Ukrainian aide whom he met in August in Madrid.

Unquestionably, Trump sought to vest Giuliani with at least some informal authority to operate on his behalf. He urged President Volodymyr Zelenskiy of Ukraine, during a phone call July 25, to consult Giuliani about the investigations he wanted. And he urged his own diplomats and aides involved with Ukraine to consult with Giuliani after they returned from Zelenskiy’s inauguration in May.

Among them was Gordon Sondland, the U.S. ambassador to the European Union, who testified that Trump instructed him to “talk to Rudy” about Ukraine and said that Giuliani had made it clear to him that he spoke for the president.

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In a letter in May from Giuliani that Parnas turned over last week to House investigators, the president’s lawyer told Zelenskiy that he was acting with Trump’s “knowledge and consent.” Parnas, who faces felony charges involving campaign finance violations, said in interviews that Ukrainian officials met with him because Giuliani assured them that he represented both him and Trump.

But virtually nothing is known about the substance of communications between Giuliani and Trump, although they appear to have spoken regularly. Mulvaney told associates that he would leave the room whenever the men would talk in order to preserve attorney-client privilege.

Did Trump explicitly link the freeze of military assistance to Zelenskiy’s public announcement of investigations?

Whether the president decided to withhold nearly $400 million in military aid to Ukraine for his own political gain, at the expense of the nation’s strategic foreign policy interests, is at the crux of the case against him. His decision thwarted the will of Congress, undercut a U.S. ally enmeshed in a war with Russia and, according to a report last week by the nonpartisan Government Accountability Office, violated U.S. law.

During the House inquiry, Sondland testified that he had informed Ukraine that it would most likely not receive the aid unless it was willing to commit to carrying out the investigations Trump wanted. But he also testified that Trump insisted to him there was no “quid pro quo” — although only after the aid freeze had become public and the president had been told about the whistleblower complaint setting out details of the pressure campaign.

Democrats in the Senate are seeking testimony from four other witnesses who played key roles in White House deliberations about the suspension in aid: Bolton; Mulvaney; Robert Blair, a senior adviser to Mulvaney; and Michael Duffey, associate director of the Office of Management and Budget.

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The New York Times reported last month that many administration officials involved in carrying out the aid freeze were kept in the dark about the president’s motivations.

Mulvaney said at a news conference in October that the aid had been withheld in part because Trump wanted an investigation into a debunked conspiracy theory that Ukraine, not Russia, had interfered in the 2016 election. Mulvaney later said that was not true and that the aid was withheld only because of concerns about Ukraine’s willingness to battle corruption and whether other nations were providing their fair share of aid to Ukraine.

In mid-August, Bolton unsuccessfully tried to persuade Trump to lift the freeze. In an Oval Office meeting later that month, Bolton, Secretary of State Mike Pompeo and Defense Secretary Mark Esper asked the president to release the funds but were rebuffed.

Duffey, a political appointee, enforced the hold on the aid after taking control of the funds from a career budget officer, Mark Sandy — a highly unusual move.

On July 25, after days of exchanges about the topic but also just 90 minutes after Trump and Zelenskiy held their fateful telephone conversation, Duffey reiterated to Defense Department officials in an email that no funds should be disbursed — instructions he said should be “closely held” because of “the sensitive nature of the request.”

How much did other top aides know about the president’s demands that Ukraine announce investigations?

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“Everyone was in the loop.”

That was Sondland’s characterization of who knew what about the push to win a commitment from the Ukrainians to announce the investigations. He testified that several top officials, including Pompeo and Mulvaney, knew that Trump would extend an Oval Office invitation to Zelenskiy only if Ukraine publicly announced the investigations.

Fiona Hill, at the time the top Russia specialist on the National Security Council, testified that at a White House meeting July 10, Sondland said that he had a deal with Mulvaney: an Oval Office invitation for Zelenskiy in exchange for a public statement that the inquiries were underway.

Although the State Department refused Sondland’s request for documents to support his testimony, he produced several emails to back up his assertions. On July 18, he wrote a group of officials — including Mulvaney, Pompeo, Bolton and Rick Perry, then the energy secretary — that Zelenskiy was ready to promise the president in their upcoming phone call that his prosecutors would “turn over every stone.”

Pompeo was among the officials who listened to the July 25 phone call in which Trump raised with Zelenskiy the need to investigate the Bidens and the 2016 election and Zelenskiy seemed to agree.

In August, Sondland wrote Pompeo that Zelenskiy would deliver a public statement that “will hopefully make the boss happy enough to authorize an invitation” because it would include “specifics.” That meant, he testified, that Zelenskiy would name Burisma, a Ukrainian company that had hired Hunter Biden, as an investigative target, along with the 2016 election.

Did Vladimir Putin and Viktor Orban sour Trump on Ukraine?

A number of State Department witnesses blamed Giuliani for Trump’s animus toward Ukraine, saying they struggled mightily to counteract his influence. But there are indications that President Vladimir Putin of Russia and Prime Minister Viktor Orban of Hungary helped solidify Trump’s views.

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Volker, the envoy to Ukraine, testified that the president’s negative opinion of Ukraine was “very deeply rooted” and evident as far back as September 2017, when Trump met Zelenskiy’s predecessor, Petro Poroshenko, in the Oval Office.

One explanation is that Trump blamed the Ukrainians for helping to expose the financial misdeeds of Paul Manafort, who was forced to resign as Trump’s campaign chairman in August 2016 and is now in prison for his crimes. In an Oval Office meeting May 23, Trump insisted to his aides that Ukrainians were “terrible people” who “tried to take me down.”

But George Kent, a deputy assistant secretary of state, testified that Trump’s view of Zelenskiy and Ukraine had darkened in the interval between that meeting and his first phone call with the Ukraine leader a month earlier, in which he congratulated him on his victory.

In the interim, Putin reportedly disparaged Zelenskiy to Trump in a phone call May 3. And Trump held an Oval Office meeting with Orban, who is antagonistic toward Zelenskiy. No transcripts of those conversations have been released. Kent attributed the shift in Trump’s attitude to the combined influence of those two foreign leaders and Giuliani.

Where were the White House lawyers

A group of government lawyers is charged with monitoring White House and National Security Council decisions for unethical or illegal behavior. The top-ranking lawyers involved in the Ukraine affair were John Eisenberg at the National Security Council and Pat Cipollone, the White House counsel.

Two Security Council staff members — Hill and Lt. Col. Alexander Vindman — told Eisenberg they feared that Sondland was improperly pressuring Ukraine to benefit Trump politically. Hill testified that Bolton had ordered her to tell Eisenberg that he did not want any part of “whatever drug deal” that Sondland and Mulvaney were cooking up.

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According to people familiar with the situation, Eisenberg shared those concerns with Cipollone, his superior, but rejected Cipollone’s advice that he bring them up with the president.

Vindman also reported concerns about the president’s July 25 phone call. Eisenberg warned him not to discuss the call with others and ordered that access be restricted to the reconstructed transcript. A person briefed on his actions said officials misinterpreted that directive as an order to put the transcript on the White House’s most secure computer.

Eisenberg eventually alerted the Justice Department to the July 25 call, but only weeks later, after the CIA’s top lawyer informed him that a CIA officer had filed an anonymous complaint about it.