RANKS OF p-CLASS GROUPS IN CYCLIC p-EXTENSIONS OF ANTI-CYCLOTOMIC Z2-EXTENSIONS [article]

Ariella Kirsch
2019
In [1], Iwasawa proved a structure theorem for the l-class group in Zl-extensions. In this thesis, we consider instead the p-class group in Zl-extensions, particularly when l=2 and p=3. Fixing K0 =Q(i), we let L0/K0 be a cyclic degree p extension and let L∞/L0 be the lift of the anti-cyclotomic Z2-extension of K0. The rank of the ambiguous ideal class group is given by Chevalley's formula. We study the question, does Chevalley's formula in fact explain the entire growth in the rank of the class
more » ... group? We study the ranks of the class groups of L0 and L1, proving results that give the rank of the class group when p = 3 and developing heuristics. We also consider the unit structure of Kn and prove that if a relative unit of Kn is the norm of an element in Ln modulo pth powers, then all of the relative units must be norms modulo pth powers. Additionally, we include computational results and evidence that Iwasawa's structure theorem does not extend to the p-class group in Zl -extensions.
doi:10.13016/1mwf-kpsx fatcat:rlz7wi4zojf4xlqgrdzwujlw6e