The lawsuit in Denver is one of several across the country arguing that former President Donald J. Trump is ineligible to hold office again.The continued existence of former President Donald J. Trump’s 2024 campaign is being litigated this week in an unassuming courtroom in Colorado.The trial stems from a lawsuit brought by voters in the state who argue that Mr. Trump is ineligible to hold office under the 14th Amendment of the Constitution because of his actions before and during the Jan. 6, 2021, attack on the Capitol. And the Colorado disqualification case isn’t isolated. Oral arguments stemming from a similar suit, in Minnesota, were held on Thursday.Here is a look at the Colorado case and beyond.What is the background on the Colorado lawsuit?It was filed in September in a state district court in Denver by six Colorado voters — four Republicans and two independents — who are suing with the help of the watchdog group Citizens for Responsibility and Ethics in Washington.These voters argue that Mr. Trump’s presence on the Republican primary ballot next year would harm them by siphoning support from their preferred candidates and, if he won the nomination, by depriving them of the ability “to vote for a qualified candidate in the general election.”They are demanding that the Colorado secretary of state not print Mr. Trump’s name on the ballot, and are asking the court to rule that Mr. Trump is disqualified in order to end any “uncertainty.”What is the 14th Amendment, and what does it say?The Colorado case specifically concerns Section 3 of the 14th Amendment, which says:No person shall be a Senator or Representative in Congress, or elector of President and Vice-President, or hold any office, civil or military, under the United States, or under any State, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any State legislature, or as an executive or judicial officer of any State, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof. But Congress may by a vote of two-thirds of each House, remove such disability.The central questions are whether the 14th Amendment applies to the presidency; whether Mr. Trump’s behavior before and on Jan. 6 constitutes “engaging in insurrection or rebellion against” the Constitution; and whether election officials or the courts can deem a person ineligible under Section 3 without specific action by Congress identifying that person.Constitutional experts have emphasized in interviews with The New York Times that the answers to these questions are not simple or self-evident.In public writings, some scholars have argued that Mr. Trump is ineligible. In an academic article, the conservative law professors William Baude and Michael Stokes Paulsen concluded: “It is unquestionably fair to say that Trump ‘engaged in’ the Jan. 6 insurrection through both his actions and his inaction.” Others have argued the opposite, with the law professors Josh Blackman and Seth Barrett Tillman saying in a recent draft paper that they see “no sound basis” for Mr. Baude’s and Mr. Paulsen’s conclusions.What is the plaintiffs’ side saying?From Monday through Wednesday, lawyers for the plaintiffs — the six Colorado voters — called seven witnesses:Daniel Hodges, a Washington, D.C., police officer, and Winston Pingeon, a Capitol Police officer, who were at the Capitol on Jan. 6. They testified that rioters had come equipped with tactical gear and had made it clear that they believed themselves to be acting on Mr. Trump’s behalf. On cross-examination, lawyers for Mr. Trump sought to distance him from the rioters, noting that the officers could not know that any individual rioter had heard his speech.Representative Eric Swalwell, Democrat of California, who said lawmakers had read Mr. Trump’s Twitter posts during the attack and saw them as connected “to our own safety in the chamber and also the integrity of the proceedings.” On cross-examination, lawyers for Mr. Trump quoted Mr. Swalwell’s own Twitter post urging Democrats to “fight” against abortion restrictions and asked if that was a call for violence; Mr. Swalwell said no.William C. Banks, a law professor at Syracuse University and an expert on presidential authority in national security. He testified that Mr. Trump could have deployed National Guard troops without a request or permission from local officials.Peter Simi, a professor of sociology at Chapman University and an expert on political extremism. He testified that the far right used “doublespeak” — language that insiders understood to be calling for violence but that maintained plausible deniability. For years, he said, Mr. Trump built credibility with members of groups like the Proud Boys and Oath Keepers, such that they saw him as an ally speaking to them in that way.Gerard Magliocca, a law professor at Indiana University and an expert on Section 3 of the 14th Amendment. He said that when the amendment was ratified, “insurrection” was understood to refer to “any public use of force or threat of force by a group of people to hinder or prevent the execution of the law,” and “engaged” meant “any voluntary act in furtherance of an insurrection, including words of incitement.”Hilary Rudy, a deputy elections director in the Colorado secretary of state’s office. She testified that the secretary of state had a legal obligation to grant ballot access only to qualified candidates, that courts could play a legitimate role in determining who was qualified, and that the office would abide by whatever the court decided.The plaintiffs’ lawyers plan to call one additional witness Friday afternoon.What is Trump’s side saying?As of Thursday, lawyers for Mr. Trump had called six witnesses:Kashyap Patel, a former chief of staff at the Defense Department. He testified that Mr. Trump had pre-emptively authorized the deployment of 10,000 to 20,000 National Guard troops to keep the peace on Jan. 6, and that they were absent because the mayor of Washington had not requested them. Under cross-examination, Mr. Patel said he did not know of any document showing Mr. Trump’s authorization.Katrina Pierson, a former spokeswoman for Mr. Trump’s campaign, who described internal disagreements over who should speak at Mr. Trump’s Jan. 6 rally. She testified that Mr. Trump nixed most of the planned speakers, including the most incendiary ones. She also said he had expressed a desire for 10,000 National Guard troops.Amy Kremer, an organizer of the Jan. 6 rally on the Ellipse, called the rally attendees “freedom-loving citizens” and “happy warriors,” and said she had seen no indication of violence or violent intent while Mr. Trump was speaking. Under cross-examination, she acknowledged that she had been inside the area that required magnetometer scans, and that she would not have seen anything that happened outside that area.Thomas Van Flein, general counsel and chief of staff to Representative Paul Gosar, Republican of Arizona. He testified that the rally crowd was peaceful, but acknowledged that he had left before Mr. Trump spoke.Tom Bjorklund, who is the treasurer of the Colorado Republican Party but testified as a private citizen, attended Mr. Trump’s speech and then went to the Capitol, where he witnessed the riot but did not enter the building himself. He said in the first part of his testimony that he had not seen any violence from Trump supporters. Later, he said he had watched people break windows, but advanced the conspiracy theory that it was a false-flag operation by “antifa.” He also said he had understood Mr. Trump’s “instructions” to be for peaceful protest.Representative Ken Buck, Republican of Colorado, testified that he believed the Jan. 6 committee’s report — which the plaintiffs have frequently cited as evidence in their case — was one-sided in its assessment of Mr. Trump’s “culpability” in the attack.Mr. Trump’s team plans to call one more witness Friday morning: an expert who will offer a different interpretation from Professor Magliocca’s of the wording in Section 3 of the 14th Amendment.What has the judge said?Before the trial began on Monday, Mr. Trump’s team made several motions to dismiss the case. Judge Sarah B. Wallace, who is overseeing the trial, rejected them.On Wednesday, after the plaintiffs had finished calling most of their witnesses, Mr. Trump’s lawyers requested a “directed verdict” — a conclusion, before the defense had called any witnesses, that no legally sufficient basis existed for the plaintiffs to prevail. They argued that even if the plaintiffs’ claims were accepted as fact, that would not legally justify disqualifying Mr. Trump. His words, they said, did not meet the Supreme Court’s standard for incitement and therefore were protected by the First Amendment.Judge Wallace denied the request, but emphasized that her denial should not be construed as a ruling on the legal questions involved — including whether Mr. Trump had “engaged in insurrection” as the 14th Amendment meant that phrase, and whether the First Amendment limited how the 14th could be applied.Rather, she said she was denying the request because in order to grant it, “I would have to decide many legal issues that I am simply not prepared to decide today.”What happens next?It is not clear how long it will take for Judge Wallace to rule after the trial ends on Friday.However, the trial is being conducted under an expedited process with the goal of having a final resolution before a January deadline for the Colorado secretary of state to certify who is on the primary ballot — and everyone involved understands that her initial ruling needs to come with enough time for appeals to be resolved, too.The United States Supreme Court is expected to have the final say.Chris Cameron More