Despite President Donald Trump’s warning that Pennsylvania’s election would be marred by “rampant and unchecked cheating,” requiring his lawyers to swiftly descend on the state, Republican lawyers brought no legal challenge raising broad questions about voting in the state by the time polls closed Tuesday.

But even amid that relative calm on Election Day, lawyers on both sides were anxiously eyeing signs that more serious legal battles could be on the horizon — particularly over how and whether certain mail ballots should be counted.

On Tuesday, GOP lawyers filed suit in state and federal courts contesting efforts by some Pennsylvania counties to allow voters an opportunity to correct mistakes in their mail ballots, such as missing signatures.

Hanging over all of those smaller-scale issues was the still lingering question of whether the U.S. Supreme Court would decide to weigh in on the hotly contested legal fight over whether Pennsylvania could count mail ballots that arrive in the next three days as long as they had been postmarked by Election Day.

“I’m proud of how Pennsylvanians conducted themselves in this historic election during a global pandemic,” Gov. Tom Wolf said in a statement Tuesday night. The Department of State, responsible for overseeing the election, said the most common issues were polling places that opened late, long lines to vote, and confusion among voters who showed up in person after asking for a mail ballot — all typical Election Day complaints.

Already, this election has been one of the most litigated in Pennsylvania’s history, with key rulings on whether mail ballots can be rejected based on signature comparisons, how long the state’s mail ballot count can continue, and fights over poll watcher access to election offices.

It was unclear Tuesday just how many mail ballots might be affected by the latest legal wrangling — especially if the Supreme Court were to intervene and decide to throw out properly postmarked votes that arrive between Tuesday and a Friday deadline established by a Pennsylvania Supreme Court ruling.

Pressed by Republicans to issue an emergency order to overturn Pennsylvania’s Friday cutoff for mail votes, the U.S. Supreme Court declined to do so last week but left open the possibility that it could revisit the issue after Election Day.

A federal judge in Washington, D.C. took a sweeping step Tuesday to reduce the number of late-arriving ballots, ordering the U.S. Postal Service to scour mail processing facilities in Philadelphia, central Pennsylvania, and 10 other locations across the country to ensure any ballots remaining there had been delivered by the end of the day. The Postal Service told the court it had done its best to comply, but that it could not guarantee it would meet the judge’s deadline.

Still, ballots had been rapidly flowing into election offices over the past several days. The Department of State said 93,000 mail ballots were returned Monday alone, and nearly 800,000 had been returned within the past week. By late afternoon Tuesday, nearly 83% of the three million mail ballots sent to voters had been returned, the department said.

But GOP lawyers zeroed in on Secretary of State Kathy Boockvar, an appointee of Wolf, alleging she had improperly advised counties that voters whose mail ballots were at risk of being disqualified — due to such failings as missing signatures and secrecy envelopes — could be contacted and given the opportunity to fix the problems or cast an emergency ballot before polls closed.

In a suit filed in state Commonwealth Court, Republican attorneys asked that impacted ballots be set aside to determine how or if they should be counted, and argued that Boockvar’s guidance was “in clear contravention” of state law. The case was scheduled for a status review Wednesday. Senate Republican leaders also called on Boockvar to resign.

Boockvar responded at a news conference Tuesday evening, saying: “We don’t think we broke the law,” adding: “I can tell you that the claims, we completely dispute.” She said she had no intentions of stepping down and blamed Republican legislative leaders for failing to authorize counties to process mail ballots before Election Day.

It was not immediately clear how many ballots the GOP lawsuit might affect statewide. But in Montgomery County, which was separately sued Tuesday over a similar issue in federal court in Philadelphia, county officials said that of all voters contacted over possible mistakes, only 49 had corrected their ballots. By Tuesday night, the county had already received nearly 240,000 mail ballots.

In that suit, Kathy Barnette — the Republican candidate running to represent Pennsylvania’s 4th Congressional District — has asked U.S. District Judge Timothy Savage to declare those corrected ballots void. Her lawyers argue that since not all counties had the staff to conduct outreach to voters to address ballot problems, efforts to correct them in Democratic-leaning Montgomery County could unfairly skew the vote. The 4th District includes a sliver of Berks County.

“Pennsylvania voters should not be treated differently based on the county where they are required to vote,” Barnette’s lawyer, Andrew Teitelman, wrote.

Montgomery County officials stood by their efforts.

“We believe our process is sound and permissible under the Election Code,” Montgomery County spokesperson Kelly Cofrancisco said.

The judge has set a hearing for Wednesday.

In Philadelphia’s election court, a Trump campaign lawyer objected to the city’s plan to post a list of disqualified mail ballots on grounds similar to those raised in the other lawsuits. But Common Pleas Judge Joshua Roberts denied her request to block the city from doing so.

Late Tuesday night, the attorney for the president’s campaign also argued that observers monitoring the activity of election workers at the Convention Center could barely see anything due to large distances between observers and election workers.

“They are not there to be potted plants,” Linda Kerns, a Trump campaign lawyer, said of the observers. “All we’re asking is if there’s things going on at the canvass, representatives can meaningfully report back to candidates what’s going on. But that is not happening."

Common Pleas Court Judge Stella Tsai ruled that the Board of Elections had complied with proper observer procedures, but also said she wouldn’t discourage the board from considering ways to make the setup more conducive for those there to watch.

Otherwise, the disputes that dominated Philadelphia’s election court docket were mostly routine, highly localized to individual polling sites, and would impact relatively few votes. The vast majority were resolved without requiring a ruling from a judge.

Beginning Wednesday, lawyers can file additional appeals of Board of Elections decisions at a docketing area to be set up at the Pennsylvania Convention Center. The process can send disputes into a courtroom to be heard by a judge, and officials expect it could last several days, or even into next week.

Andrew Wellbrock, head of the election task force at the Philadelphia District Attorney’s Office, said voting Tuesday appeared to have run so smoothly at Philadelphia’s polling places that the biggest challenge they faced all day was debunking disinformation shared on social media — in some cases, by Trump campaign staffers.

By Tuesday night, city prosecutors had received 68 complaints, resolving virtually all.

But Wellbrock said the task force received nearly as many calls from people around the country “screaming at us on the phone” and alleging they had seen videos of votes being stolen, or that the DA’s Office was not taking election interference seriously — allegations that Wellbrock said were unsupported by evidence.

By and large, he said, despite the anxiety about voter intimidation and interference at the polls, Election Day in Philadelphia ran no differently than it had in past elections.

And on that point, even the chair of Philadelphia’s Republican City Committee, State Rep. Martina White, appeared to agree.

She said the GOP had a team of attorneys roaming Philadelphia in search of issues at polling sites. Asked midday whether they’d identified any significant problems, she said she did not know of any.

Staff writer Chris Brennan contributed to this article.