Computations in non-commutative Iwasawa theory

Root numbers, Selmer groups, and non-commutative Iwasawa theory

Global root numbers have played an important role in the study of rational points on abelian varieties since the discovery of the conjecture of Birch and Swinnerton-Dyer. The aim of this paper is to throw some new light on this intriguing and still largely conjectural relationship. The simplest avatar of this phenomenon is the parity conjecture which asserts that for an abelian variety A over a...

Non-commutative Polynomial Computations

In [6] Mora described an algorithm which, if it terminates, returns a noncommutative Gröbner basis. Here we follow that paper to prove the correctness of the algorithm as implemented by Dié Gijsbers and the author in GAP, cf. [1]. The algorithm is the core algorithm of the GAP package GBNP for computing with non-commutative polynomials. Earlier versions of this section were written with the hel...

Dimension Computations in Non-Commutative Associative Algebras

Computations About Tate-Shafarevich Groups Using Iwasawa Theory

We explain how to combine deep results from Iwasawa theory with explicit computation to obtain information about p-parts of Tate-Shafarevich groups of elliptic curves over Q. This method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin and Kato do not apply.

IWASAWA THEORY OF TOTALLY REAL FIELDS FOR CERTAIN NON-COMMUTATIVE p-EXTENSIONS

In this paper, we prove the Iwasawa main conjecture (in the sense of [CFKSV]) of totally real fields for certain specific noncommutative p-adic Lie extensions, using the integral logarithms introduced by Oliver and Taylor. Our result gives certain generalization of Kato’s proof of the main conjecture for Galois extensions of Heisenberg type ([Kato1]). 0. Introduction 0.