Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg.
The first conjecture was proposed in 1976 and concerns Iwasawa invariants.
The conjecture, also referred to as Greenberg's invariants conjecture, firstly appeared in Greenberg's Princeton University thesis of 1971 and originally stated that, assuming that
is a totally real number field and that
dividing the class number of
Note that if Leopoldt's conjecture holds for
is the cyclotomic one (since it is totally real).
In 1976, Greenberg expanded the conjecture by providing more examples for it and slightly reformulated it as follows: given that
is a fixed prime, with consideration of subfields of cyclotomic extensions of
, one can define a tower of number fields
is totally real, is the power of
dividing the class number of
is an arbitrary number field, then there exist integers
dividing the class number of
λ = μ = 0
Simply speaking, the conjecture asks whether we have
for any totally real number field
, or the conjecture can also be reformulated as asking whether both invariants λ and μ associated to the cyclotomic
-extension of a totally real number field vanish.
In 2001, Greenberg generalized the conjecture (thus making it known as Greenberg's pseudo-null conjecture or, sometimes, as Greenberg's generalized conjecture): Supposing that
is a totally real number field and that
(Recall that if Leopoldt's conjecture holds for
Hilbert class field of
, regarded as a module over the ring
Another related conjecture (also unsolved as of yet) exists: We have
This related conjecture was justified by Bruce Ferrero and Larry Washington, both of whom proved (see: Ferrero–Washington theorem) that
of the rational number field
Another conjecture, which can be referred to as Greenberg's conjecture, was proposed by Greenberg in 2016, and is known as Greenberg's
It states that for any odd prime
This conjecture is related to the Inverse Galois problem.