Jack Smith, the special counsel prosecuting former President Donald J. Trump on charges of plotting to overturn the 2020 election, urged the Supreme Court on Monday to reject Mr. Trump’s claim that he is immune from prosecution.

“The president’s constitutional duty to take care that the laws be faithfully executed does not entail a general right to violate them,” Mr. Smith wrote.

The filing was Mr. Smith’s main submission in the case, which will be argued on April 25.

He wrote that the novelty of the case underscored its gravity.

“The absence of any prosecutions of former presidents until this case does not reflect the understanding that presidents are immune from criminal liability,” Mr. Smith wrote. “It instead underscores the unprecedented nature of petitioner’s alleged conduct.”

He urged the justices not to lose sight of the basic legal terrain.

“A bedrock principle of our constitutional order,” he wrote, “is that no person is above the law — including the president.” He added, “The Constitution does not give a president the power to conspire to defraud the United States in the certification of presidential-election results, obstruct proceedings for doing so or deprive voters of the effect of their votes.”

Mr. Smith urged the court to move quickly, though he did not directly address the pending election.

When the Supreme Court said in February that it would hear the case, it set what it called an expedited schedule. But it was not particularly fast, with oral arguments scheduled about seven weeks later. That delay was a significant partial victory for Mr. Trump, whose trial had been expected to start March 4.