Others are less convinced by Mr. Kobach’s claims that voter fraud is pervasive and widespread. “To me, it’s a fix to a problem that doesn’t exist,” said Jeff Nielsen, standing behind the counter of a pawnshop he manages in Olathe, Kan. Mr. Nielsen, who works in Kansas, lives across the border in Missouri, grew up Republican and supports gun ownership — a wall of rifles was behind him in the shop — but now votes for Democrats. “I think Kobach should be a little more concerned with Russians picking our president than with Mexicans voting,” he said.

In 2016, Federal District Judge Julie A. Robinson in Kansas ruled that the proof-of-citizenship requirement of the state law that Mr. Kobach championed violated federal law. If Judge Robinson rules against Mr. Kobach in the current trial, thousands of Kansans who attempted to register to vote at a Department of Motor Vehicles office but did not provide proof of citizenship will see their registrations reinstated. If Mr. Kobach prevails, it could become more difficult for residents without proof of citizenship to vote in the 2018 elections.

Early in the trial, Judge Robinson reprimanded Mr. Kobach, who frequently stumbled on legal procedure, for attempting to display documents that had not been formally introduced as evidence. “Evidence 101 — not going to do it,” she said.

In his opening statements, Mr. Kobach, Harvard-educated and Yale Law School-trained, criticized current rules as weak and porous, arguing that a requirement that voters attest on a form at the Department of Motor Vehicles that they are citizens — risking a penalty of perjury for a false statement — is not enough. “All you had to do was check a box,” he said. “That was essentially it.”

Mr. Kobach says that even though he has documented a relatively small number of noncitizens in Kansas who have attempted to vote, he has reason to believe that they are only “the tip of the iceberg” and that thousands more exist on the state voter rolls.

But Dale Ho, a lawyer for the A.C.L.U., said that the law Mr. Kobach lobbied for has made it difficult or impossible for some state residents who do not have a birth certificate, passport or other proof of citizenship to vote. Experts in election law say that noncitizens successfully voting rarely happens.