1. CMB 2011 (vol 56 pp. 148)
|On the Gras Conjecture for Imaginary Quadratic Fields|
In this paper we extend K. Rubin's methods to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field $k$ and prime numbers $p$ that divide the number of roots of unity in $k$.
Keywords:elliptic units, Stark units, Gras conjecture, Euler systems
Categories:11R27, 11R29, 11G16
2. CMB 2007 (vol 50 pp. 334)
|Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields |
We obtain Hauptmoduls of genus zero congruence subgroups of the type $\Gamma_0^+(p):=\linebreak\Gamma_0(p)+w_p$, where $p$ is a prime and $w_p$ is the Atkin--Lehner involution. We then use the Hauptmoduls, along with modular functions on $\Gamma_1(p)$ to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.
Categories:11F03, 11G16, 11R20