Mathematics Colloquium

The study of Diophantine equations has occupied number theorists for centuries and led to Kummer's introduction of the class number. Nearly a century ago, Herbrand refined Kummer's criterion for the class numbers of cyclotomic fields; the converse was established in 1976 by Ribet, who introduced a modular approach. In the same year, Mazur used modular methods to investigate rational torsion points on elliptic curves. The impact of these groundbreaking results continues to shape modern research in number theory. In this talk, we review the key ideas underlying these methods, emphasizing the role of the Eisenstein ideal, and conclude with a brief discussion of ongoing work.