Greenberg's conjectures

Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg. Both are still unsolved as of 2021.

Invariants conjecture
The first conjecture was proposed in 1976 and concerns Iwasawa invariants. This conjecture is related to Vandiver's conjecture, Leopoldt's conjecture, Birch–Tate conjecture, all of which are also unsolved.

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  is the cyclotomic -extension, , i.e. the power of  dividing the class number of  is bounded as . Note that if Leopoldt's conjecture holds for  and , the only -extension of  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 finite extension of  and that  is a fixed prime, with consideration of subfields of cyclotomic extensions of , one can define a tower of number fields
 such that  is a cyclic extension of  of degree . If  is totally real, is the power of  dividing the class number of  bounded as  ? Now, if  is an arbitrary number field, then there exist integers ,  and  such that the power of  dividing the class number of  is , where  for all sufficiently large . The integers , ,  depend only on  and . Then, we ask: is  for  totally real?

Simply speaking, the conjecture asks whether we have  for any totally real number field  and any prime number , 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  is a prime, let  denote the compositum of all -extensions of . (Recall that if Leopoldt's conjecture holds for  and , then .) Let  denote the pro- Hilbert class field of  and let , regarded as a module over the ring . Then  is a pseudo-null -module.

A possible reformulation: Let  be the compositum of all the -extensions of  and let , then  is a pseudo-null -module.
 
Another related conjecture (also unsolved as of yet) exists:

We have  for any number field  and any prime number .

This related conjecture was justified by Bruce Ferrero and Larry Washington, both of whom proved (see: Ferrero–Washington theorem) that  for any abelian extension  of the rational number field  and any prime number .

p-rationality conjecture
Another conjecture, which can be referred to as Greenberg's conjecture, was proposed by Greenberg in 2016, and is known as Greenberg's -rationality conjecture. It states that for any odd prime  and for any , there exists a -rational field  such that . This conjecture is related to the Inverse Galois problem.

Further reading
R. Greenberg, On some questions concerning the lwasawa invariants, Princeton University thesis (1971)
R. Greenberg, "On the lwasawa invariants of totally real number fields", American Journal of Mathematics, issue 98 (1976), pp. 263–284
R. Greenberg, "Iwasawa Theory — Past and Present", Advanced Studies in Pure Mathematics, issue 30 (2001), pp. 335–385
R. Greenberg, "Galois representations with open image", Annales mathématiques du Québec, volume 40, number 1 (2016), pp. 83–119
B. Ferrero and L. C. Washington, "The Iwasawa Invariant  Vanishes for Abelian Number Fields", Annals of Mathematics (Second Series), volume 109, number 2 (May, 1979), pp. 377–395

Algebraic number theory
Conjectures
Unsolved problems in number theory