GLOBAL UNITS MODULO ELLIPTIC UNITS AND IDEAL CLASS GROUPS

Global Units modulo Circular Units : descent without Iwasawa’s Main Conjecture

Iwasawa’s classical asymptotical formula relates the orders of the p-parts Xn of the ideal class groups along a Zp-extension F∞/F of a number field F , to Iwasawa structural invariants λ and μ attached to the inverse limit X∞ = lim ← Xn. It relies on ”good” descent properties satisfied by Xn. If F is abelian and F∞ is cyclotomic it is known that the p-parts of the orders of the global units mod...

Class Groups, Totally Positive Units, and Squares

Given a totally real algebraic number field K, we investigate when totally positive units, U¿, are squares, u£. In particular, we prove that the rank of U¿ /Ují is bounded above by the minimum of (1) the 2-rank of the narrow class group of K and (2) the rank of Ul /U¿ as L ranges over all (finite) totally real extension fields of K. Several applications are also provided. 1. Notation and prelim...

O ct 2 00 6 Global Units modulo Circular Units : descent without Iwasawa ’ s Main Conjecture . ∗

Iwasawa’s classical asymptotical formula relates the orders of the p-parts Xn of the ideal class groups along a Zp-extension F∞/F of a number field F , to Iwasawa structural invariants λ and μ attached to the inverse limit X∞ = lim ← Xn. It relies on ”good” descent properties satisfied by Xn. If F is abelian and F∞ is cyclotomic it is known that the p-parts of the orders of the global units mod...

Special units and ideal class groups of extensions of imaginary quadratic fields

Modular Units and Cuspidal Divisor Class Groups of X1(n)

Abstract. In this article, we consider the group F∞ 1 (N) of modular units on X1(N) that have divisors supported on the cusps lying over∞ of X0(N), called the ∞-cusps. For each positive integer N , we will give an explicit basis for the group F∞ 1 (N). This enables us to compute the group structure of the rational torsion subgroup C∞ 1 (N) of the Jacobian J1(N) of X1(N) generated by the differe...