Predicative Unit Classes (Sets) in Text Design

The enormity and comprehensiveness of predication alongside its ordinariness constantly encountered in the usual flow of speech significantly complicate any practical implementation of it in SLA and textbook theory. The phenomenon being elucidated predominantly in abstract terms widens the gap between its theoretical exploration and practical application in the field mentioned. There emerges a necessity of the search for a new notional direction from which the point is viewed. The new frame of reference would have far reaching implications for language acquisition and textbook development providing a strong theoretical underpinning for language curricula. Invariant binary predicative units are described as sense building blocks of predication. Their dichotomically organized classes are presented in this Chapter. Class 1 dichotomizes full predicative units and elliptic units. Class 2 includes units with substituted arguments or predicates. Class 3 comprises objective (semantic) constants and subjective (individual) variables. Class 4 takes in units with modificative and propositional predicates. Class 5 is composed of units containing analytic and synthetic predicates. Binary predicative units of the types categorized in the classes act conjointly, generating a unified multi-channel network of sense formation.

Annihilators of the Ideal Class Group of a Cyclic Extension of an Imaginary Quadratic Field

The aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group, and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.

Triviality of the ℓ-class groups in -extensions of for split primes p ≡ 1 modulo 4

In this paper we study the class numbers in the finite layers of certain non-cyclotomic $\mathbb{Z}$p-extensions of the imaginary quadratic field $\mathbb{Q}(\sqrt{-1})$, for all primes p ≡ 1 modulo 4. By studying the Mahler measure of elliptic units, we are able to show that there are only finitely many primes ℓ congruent to a primitive root modulo p2 that divide any of the class numbers in the $\mathbb{Z}$p-extension.

On the units generated by Weierstrass forms

There is an algorithm of Schoof for finding divisors of class numbers of real cyclotomic fields of prime conductor. In this paper we introduce an improvement of the elliptic analogue of this algorithm by using a subgroup of elliptic units given by Weierstrass forms. These elliptic units which can be expressed in terms of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x$-coordinates of points on elliptic curves enable us to use the fast arithmetic of elliptic curves over finite fields.

Class number calculation using Siegel functions

We propose a fast method of calculating the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$-part of the class numbers in certain non-cyclotomic $\mathbb{Z}_p$-extensions of an imaginary quadratic field using elliptic units constructed by Siegel functions. We carried out practical calculations for $p=3$ and determined $\lambda $-invariants of such $\mathbb{Z}_3$-extensions which were not known in our previous paper.

GLOBAL UNITS MODULO ELLIPTIC UNITS AND IDEAL CLASS GROUPS

Let p be a prime number, and let k be an imaginary quadratic number field in which p decomposes into two distinct primes 𝔭 and [Formula: see text]. Let k∞ be the unique ℤp-extension of k which is unramified outside of 𝔭, and let K∞ be a finite extension of k∞, abelian over k. Following closely the ideas of Belliard in [1], we prove that in K∞, the projective limit of the p-class group and the projective limit of units modulo elliptic units share the same μ-invariant and the same λ-invariant. We deduce that a version of the classical main conjecture, which is known to be true for p ∉ {2, 3}, holds also for p ∈ {2, 3} once we neglect the μ-invariants.