On mirror equalities and certain weak forms of Greenberg's conjecture
Let \(p\) be an odd rational prime, \(K\) a fixed number field and \(\lambda = \lambda _ p (K)\) the Iwasawa \(\lambda\) invariant associated to the cyclotomic \({\mathbb Z}_ p\)-extension \(K_ \infty/K\). When \(K\) is a CM-field containing a primitive \(p\)-th root of unity \(\zeta _ p\), we consider the Iwasawa \(\lambda\) invariant \(\lambda ^ +=\lambda _ p (K^ +)\) of the maximal totally real subfield \(K^ +\) of \(K\). Iwasawa theory yields \(\lambda ^ + \leq \lambda\). Then \(\lambda=\lambda ^ + + \lambda ^ -\) for some non-negative integer \(\lambda ^ -\). Class field theory and Kummer theory show the well known mirror inequality \(\lambda ^ + \leq \lambda ^ -\). Ralph Greenberg conjectured that \(\lambda ^ + = 0\). In particular Greenberg's conjecture predicts that if \(\lambda ^ ->0\) then \(\lambda ^ + <\lambda ^ -\). This justifies to study a weak form of Greenberg's conjecture, namely: \(\lambda ^ + = \lambda ^ -\) iff \(\lambda ^ + =0 = \lambda ^ -\). This weak form of Greenberg's conjecture is considered in this paper. The authors study only number fields \(K\) such that both Leopoldt conjecture at \(p\) and the nullity of the Iwasawa \(\mu\) invariant hold. First, using some Iwasawa modules, conditions equivalent to the condition \(\lambda ^ +=\lambda ^ -\) are found and in particular to the conditions \(\lambda ^ + =0\) and \(\lambda ^ + =\lambda ^ - =0\). The authors apply these conditions to the cases when \(K^ +\) admits only one \(p\)-place which is totally ramified in \(K_ \infty ^ +\) and when \(p\) splits in \(K ^ +\). Along the paper it is studied the same conjecture for the invariants \(\lambda (\chi)\) and \(\lambda(\chi^ \ast)\) where \(\chi\) is a \(p\)-adic character of \(K\), \(\chi^ \ast = \omega \chi ^ {-1}\) is the mirror character, \(\omega\) being the Teichmüller character and \(K\) is a totally real number field.
scientific article; zbMATH DE number 2156129
scientific article