Remarks on connections between the Leopoldt conjecture, p-class groups and unit groups of algebraic number fields (Q914744)

Let k be an algebraic number field with unit group \(E'\) and let \(\delta_ p\) denote the difference between the \({\mathbb{Z}}\)-rank and the p- adic rank of \(E'\). Thus the Leopoldt conjecture is true for k iff \(\delta_ p=0\) for each prime p. Let E consist of elements of \(E'\) which are p-th powers at every infinite place and let \(E_ S\) denote the closure of E in the product of the local unit groups taken over \({\mathfrak p}\in S\), where S is a finite set of finite places of k containing the places lying above p. Let C be the ideal class group of \(k(\zeta_ p)\), and let D be the subgroup generated by all the extensions of the ideals in S. Put \(C_ S=C/D\cdot C^ p\) considered as a \({\mathbb{Z}}_ p[Gal(k(\zeta_ p)/k)]-module,\) and let \(C_{S,\omega}\) be an eigenspace corresponding to a naturally defined idempotent. For any abelian group A, let \(t_ p(A)\) denote the subgroup of \(p^ n\)-torsion points \((n=1,2,...).\) The purpose of the paper is to study \(\delta_ p\) in connection with \(t_ p(E_ S)\) and \(C_{S,\omega}\), and to obtain sufficient conditions for \(\delta_ p=0\). By way of example we state here the first main result: We have \(\delta_ p=0\) if and only if there is a set S such that: (1) \(C_{S,\omega}\) vanishes. (2) The p-ranks of \(t_ p(E_ S)\) and \(t_ p(E)\) are equal. \((3)\quad E^ p_ S\cap E^{'p}\subseteq E^ p.\) The other results are too complicated to be reproduced here.