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 2021.

Invariants conjecture

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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

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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

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  • 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