T-ranks of Iwasawa modules

Abstract

Let p be a rational prime. We study Galois (co-)invariants of Iwasawa modules attached to Zp-extensions of number fields, by encoding the corresponding orders in so-called T-ranks. We show that the growth of T-ranks at small layers of a Zp-extension bounds the over-all growth. This allows for effective algorithms for checking whether the T-ranks of the layers in a Zp-extension remain bounded. We apply this theoretical tool for checking the boundedness of T-ranks in order to prove a conjecture of Gross for all number fields containing exactly two primes above p, and we verify this conjecture by computations for many cubic non-normal number fields with three such primes. Moreover, our method can also be used to verify numerically Leopoldt's Conjecture for K and p, provided that p is totally split in K. Finally, we study the problem of semi-simplicity.