Fitting ideals of p-ramified Iwasawa modules over totally real fields

We completely calculate the Fitting ideal of the classical p-ramified Iwasawa module for any abelian extension K/k of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results where we had to assume that only p-adic places may ramify in K/k. One of the important ingredients is the computation of some complexes in appropriate derived categories.

ON THE SELMER GROUP OF A CERTAIN -ADIC LIE EXTENSION

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication. Let $p\geq 5$ be a prime in $\mathbb{Q}$ and suppose that $E$ has good ordinary reduction at $p$. We study the dual Selmer group of $E$ over the compositum of the $\text{GL}_{2}$ extension and the anticyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic extension as an Iwasawa module.

On Greenberg’s generalized conjecture for complex cubic fields

Let [Formula: see text] be a prime number. For a number field [Formula: see text], let [Formula: see text] be the compositum of all [Formula: see text]-extensions of [Formula: see text]. Then Greenberg’s generalized conjecture (GGC) claims that the unramified Iwasawa module [Formula: see text] is pseudo-null over the Iwasawa algebra associated to the Galois group of [Formula: see text]. In this paper, we establish sufficient conditions of GGC when [Formula: see text] is a complex cubic field and give many examples which satisfy the conditions with the help of computer programs.

Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction

This paper studies the compact p∞-Selmer Iwasawa module X(E/F∞) of an elliptic curve E over a False Tate curve extension F∞, where E is defined over ℚ, having multiplicative reduction at the odd prime p. We investigate the p∞-Selmer rank of E over intermediate fields and give the best lower bound of its growth under certain parity assumption on X(E/F∞), assuming this Iwasawa module satisfies the H(G)-Conjecture proposed by Coates–Fukaya–Kato–Sujatha–Venjakob.