Greenberg's conjectures

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Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg. Both are still unsolved as of 2018.

Invariants conjecture[edit]

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 cyclomtomic 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 . 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[edit]

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 that states that for any odd prime and for any , there exist a -rational field such that . This conjecture is related to Inverse Galois problem.

Further reading[edit]

  • 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