When the deputy sheriff’s patrol cruiser pulled up beside him as he walked down Broad Street at sunset last August, Martee Flournoy, a 32-year-old black man, was both confused and rattled. He had reason: In this corner of rural Georgia, African-Americans are arrested at a rate far higher than that of whites. But the deputy had not come to arrest Mr. Flournoy. Rather, he had come to challenge Mr. Flournoy’s right to vote. The majority-white Hancock County Board of Elections and Registration was systematically questioning the registrations of more than 180 black Sparta citizens — a fifth of the city’s registered voters — by dispatching deputies with summonses commanding them to appear in person to prove their residence or lose their voting rights. “When I read that letter, I was kind of nervous,” Mr. Flournoy said in an interview. “I didn’t know what to do.”