Albuquerque has emerged as a flash point for people from the United States and Latin America who are pursuing applications, thanks in part to the expertise of historians such as Ms. Koplik in certifying Sephardic heritage, and the openness of the state’s Jewish Federation to examining applications from people who are not practicing Jews. Hundreds of applicants have already traveled here to take language and civics exams at the Cervantes Institute, an organization funded by Spain’s government to promote the teaching of the Spanish language and culture.

Ms. Koplik said that applicants have come from around the Americas, but are largely divided into three groups: Venezuelans trying to flee their country’s severe economic crisis; Mexicans from the relatively prosperous state of Nuevo León, where there is a large concentration of people with documented Sephardic ancestry; and multigenerational Hispanic families with roots in what is now the American Southwest.

The Spanish law allows applicants to pursue citizenship by proving that they have at least one Sephardic ancestor who fled Spain some 500 years ago. New Mexico, with its wealth of colonial-era archives and United States census data after the American conquest in 1848, stands out for its relative ease of delving into records compared with other places where so-called crypto-Jews settled.

“We know that various people who came to New Mexico in the earliest phases of Spanish colonization had Sephardic backgrounds,” said Dennis Maez, 60, a professional genealogist in Albuquerque who has already conducted detailed ancestral studies for more than 60 people applying for Spanish citizenship. “From there, it’s a matter of connecting the dots through the centuries.”

Altogether, more than 6,400 people from around the world with Sephardic ancestry have obtained Spanish citizenship under the law since 2015, including hundreds so far from the United States. Authorities in Spain this year extended the deadline for applying under the measure by a year, until October 2019, in an effort to give some applicants more time to prepare for exams and prepare vetted genealogies.

Historians have documented in recent decades how crypto-Jews converted to Catholicism under threat of death during the Spanish Inquisition but stealthily maintained Jewish practices and rituals. They moved from the Iberian Peninsula to different parts of the Spanish Empire, initially the Canary Islands, followed by colonial holdings in the Caribbean, central and northeast Mexico. By the 16th and 17th century, many had settled in New Mexico.