Ralph Greenberg phublished in 1977 his thesis, written under the direction of Kenkichi Iwasawa, in an _American Journal of Math_ paper entitled _Iwasawa invariants of totally real fields_. Back then, the so-called _Main Conjecture_ was still open and Greenberg studied the vanishing of a certain invariant $\lambda_p$ partly for its own sake and partly because in certain cases it would imply the Main Conjecture. Little by little, people start to refer at Greenberg's condition $\lambda_p=0$ for totally real base fields as "Greenberg's conjecture" although he had not conjectured anything and simply studied the question.

The Main Conjecture was proven by Mazur--Wiles (1984), then reproven in greater generality by Wiles (1990), eventually proven a third time by Kolyvagin--Rubin (1992 or so) with much easier techniques. But Greenberg's question whether $\lambda_p=0$ is still alive and active, people are trying to prove it in general and the few results contained in the original thesis, albeit somehow interesting, got more or less forgotten.