In his seminal Eisenstein ideal paper (on which there is a 50th anniversary conference next year) Mazur asked about the ranks of Eisenstein Hecke algebras of prime level and gave some data. There wasn’t much progress over the next 20 years. Then Merel in the mid-90s, Calegari-Emerton a decade later, Wake-(Wang-Erickson) and Schaefer-Stubley over the next decade were able to make some limited progress.

I’ll survey the problem and it’s history.