Higher codimension behavior in equivariant Iwasawa theory for CM-fields

Abstract

In classical Iwasawa theory, we mainly study codimension one behavior of arithmetic modules. Relatively recently, F.

Higher codimension Iwasawa theory for elliptic curves with supersingular reduction

“…The structure of E 1 (X ǫ S ) will be the theme of Theorem 1.3 below. Actually, by combining Theorems 1.1 and 1.3, we obtain an analogue of [8,Theorem 5.3] for classical Iwasawa modules for CM-fields. (The statement of the results in this paper is much simpler than that of [8]; this is essentially because we have H 0 (K ∞,p , E[p ∞ ]) = 0 for any p-adic prime p of F .…”

“…As in [8, §3.1], we define S ram,p (K ∞ /F ) as the set of finite primes v of F such that v is not lying above p and that the ramification index of K ∞ /F at v is divisible by p. For instance, we have S ram,p (K ∞ /F ) = ∅ as long as Gal(K ∞ /F ) does not contain an element of order p (this case will be called the non-equivariant case). The set S ram,p (K ∞ /F ) corresponds to the set Φ K/F in [5, Theorem 1], and the necessity of the set in our study is also explained in [8,Proposition 3.1]. We take a finite set S of primes of F such that (2.1)…”

“…In [8], the author proposed a new approach to the theory of [2] and [3]. The new approach enables us to deal with equivariant situations and, moreover, to avoid localizing at prime ideals of height 2.…”

“…In the present paper, by applying the approach of [8], we develop higher codimension Iwasawa theory for elliptic curves with supersingular reduction. This work generalizes the results of [16] in the following aspects:…”

“…The basic idea of this paper is the same as in [8]. However, we need a couple of extra ingredients which are specific to elliptic curves with supersingular reduction.…”