On the group of modular units and the ideal class group

Abstract

J.-M. Kim, S. Bae and I.-S. Lee showed that there exists an isomorphism between the p-primary part of the ideal class group and p-primary part of the unit group modulo cyclotomic unit group in Q(ζ p n ) + for all sufficiently large n under some conditions. In the present paper, we shall give an analogue of their result for modular units.

Remark on elliptic units in a $\mathbb {Z}_p$-extension of an imaginary quadratic field

“…We mention that a similar result is already given in [7] for the case that p ≥ 5 splits in k, k n is the ray class field of k modulo p n+1 , and p does not split in the absolute class field of k. (When k is a real abelian field and K/k is the cyclotomic Z p -extension, similar results are previously known. See [11], [15], etc.…”

“…In this section, we will define the group Φ n of elliptic units in k n . Our construction is similar to that of [7]. We use the same notation as given in Oukhaba [14].…”

“…[11], [15], [7], etc.). Fix a positive integer n which satisfies B l ∼ = B l and |A l | = |A n | for all l ≥ n. We can take a positive integer m > n which satisfies…”

“…The author also would like to express his gratitude to Professor Keiichi Komatsu for giving useful suggestions during the collaborative work on the previous paper [7].…”