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 F {\displaystyle F} is a totally real number field and that F ∞ / F {\displaystyle F_{\infty }/F} is the cyclotomic Z p {\displaystyle \mathbb {Z} _{p}} -extension, λ ( F ∞ / F ) = μ ( F ∞ / F ) = 0 {\displaystyle \lambda (F_{\infty }/F)=\mu (F_{\infty }/F)=0} , i.e. the power of p {\displaystyle p} dividing the class number of F n {\displaystyle F_{n}} is bounded as n → ∞ {\displaystyle n\rightarrow \infty } . Note that if Leopoldt's conjecture holds for F {\displaystyle F} and p {\displaystyle p} , the only Z p {\displaystyle \mathbb {Z} _{p}} -extension of F {\displaystyle F} 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 k {\displaystyle k} is a finite extension of Q {\displaystyle \mathbf {Q} } and that ℓ {\displaystyle \ell } is a fixed prime, with consideration of subfields of cyclotomic extensions of k {\displaystyle k} , one can define a tower of number fields
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