Let $p$ be an odd prime. We attach appropriate signed Selmer groups to elliptic curve $E$, where $E$ is assumed have semistable reduction at all primes above $p$. then compare the Iwasawa $\lambda $-invariants of these for tw

The purpose of this paper is to describe certain natural 4-vector fields on quaternionic flag manifolds, which geometrically determines the Bruhat cell decomposition. This structure naturally descends from the symplectic group, where it is related to the dressing action given by the Iwasawa decomposition of the general linear group over the quaternions.

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.

The purpose of this paper if to describe a natural 4-vector field on the quaternionic flag manifolds, which geometrically determines the Bruhat cell decomposition. This structure naturally descends from the symplectic group, where it is related to the dressing action defined via the Iwasawa decomposition of the general linear group over the quaternions.

After a too-brief introduction to adeles and ideles, we sketch proof of analytic continuation and functional equation of Riemann’s zeta, in the modern form due independently to Iwasawa and Tate about 1950. The sketch is repeated for Dedekind zeta functions of number fields, noting some additional complications. The sketch is repeated again for Hecke’s (größencharakter) L-functions, noting furth...

We establish formulae for the Iwasawa invariants of Mazur–Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of “medium” weight, and our second deals with forms of small slope. We give examples illustrating the strange behavior which can occur in the high weight, high slope case.

The existence of infinitely many elliptic curves with no rational points except the origin oo is proved by refining a theorem of DavenportHeilbronn. The existence of infinitely many quadratic fields with the Iwasawa invariant A3 = 0 is proved at the same time.

For certain cyclic cubic fields k, we verified that Iwasawa invariants λ3(k) vanished by calculating units of abelian number field of degree 27. Our method is based on the explicit representation of a system of cyclotomic units of those fields.

We give a brief exposition of the Iwasawa theory of cyclotomic extensions, so as to discuss its relationship with p-adic zeta functions. We give an overview of these connections and the arithmetic significance of the theory, leading up to a statement of the main conjecture.

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Zpextension D∞ of the CM field K, where p is a prime of good, supersingular reduction for E. Our main result yields an asymptotic formula for the corank of the p-primary Selmer group of E along the extension D∞/K.