The aim of this paper is to compare two modules of elliptic units, which arise in the study of elliptic curves E over quadratic imaginary fields K with complex multiplication by ${}_K$, good ordinary reduction above a split prime p and prime power conductor (over K). One of the modules is a special case of those modules of elliptic units studied by K. Rubin in his paper [Invent. Math. 103 (1991)] on the two-variable main conjecture (without p-adic L-functions), and the other module is a smaller one,...

This paper is devoted to a construction of new annihilators of the ideal class group of a tamely ramified compositum of quadratic fields. These annihilators are produced by a modified Rubin’s machinery. The aim of this modification is to give a stronger annihilation statement for this specific type of fields.

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic ${\mathbf{Z}}_p$-extension of the CM field, where $p$ is a prime of good, ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg show that it is not a torsion module. In...

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order $q$-recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which...

One can define class invariants for a quartic primitive CM field $K$ as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to $K$. Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct $S$-units in certain abelian extensions of a reflex field of $K$, where $S$ is effectively determined by $K$, and to bound the primes appearing...

From complex multiplication we know that elliptic units are contained in certain ray class fields over a quadratic imaginary number field $K$, and Ramachandra [3] has shown that these ray class fields can even be generated by elliptic units. However the generators constructed by Ramachandra involve very complicated products of high powers of singular values of the Klein form defined below and singular values of the discriminant $\Delta$. It is the aim of this paper to show, that in many cases a generator...

Nous étudions la structure de certains espaces homogènes principaux associés aux éléments du groupe de Selmer d’une courbe elliptique à multiplication complexe. Nous utilisons des résultats de Rubin pour construire, à partir des unités elliptiques, des espaces homogènes principaux de structure galoisienne non triviale. Cette construction fournit un lien nouveau entre un problème de structure galoisienne et certaines fonctions $L-p$-adiques.

We describe here two sets of generators of an ideal $\Delta \left(\psi \right)=M\left(\psi \right)$, of finite index inside the square $I^2$ of the augmentation ideal $I$ of $\mathbb{Z}\left[G\right]$, associated to the Dirichlet character $\psi$ of the finite group $G$. That peculiar ideal first appeared in questions related to the computation of class number formulas for abelian non ramified extensions of $𝒜$-fields cf. [2] and [3], satisfying certain special conditions which are outlined in the introduction of [1]. A rough idea of these formulas is given in §§2 and 6.

Nous comparons le comportement dans les ${\mathbb{Z}}_p$-extensions du nombre de classes d’idéaux avec le comportement de l’indice du groupe des unités elliptiques de Rubin.

Let $p$ be a prime number, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two primes $𝔭$ and $\overline{𝔭}$. Let $k_{\infty }$ be the unique ${\mathbb{Z}}_p$-extension of $k$ which is unramified outside of $𝔭$, and let $K_{\infty }$ be a finite extension of $k_{\infty }$, abelian over $k$. Let ${𝒰}_{\infty }/{𝒞}_{\infty }$ be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of ${𝒰}_{\infty }/{𝒞}_{\infty }$ are finite. Our approach uses distributions and the $p$-adic $\mathrm{L}$-function, as defined in [5].

La conjecture de Birch et Swinnerton-Dyer prédit que l’ordre $r_{\infty }$ du zéro en $s=1$ de la fonction $L$ d’une courbe elliptique $E$ définie sur $𝐐$ est égal au rang $r$ du groupe de ses points rationnels. On sait démontrer cette conjecture si $r_{\infty }=0$ ou $1$, mais on n’a aucun résultat reliant $r_{\infty }$ et $r$ si $r_{\infty }\ge 2$. Nous expliquerons comment Kato démontre que la fonction $L$$p$-adique attachée à $E$ a, en $s=1$, un...

In the previous paper [Sch2] it has been shown that ray class fields over quadratic imaginary number fields can be generated by simple products of singular values of the Klein form defined below. In the present article the second named author has constructed more general products that are contained in ray class fields thereby correcting Theorem 2 of [Sch2]. An algorithm for the computation of the algebraic equations of the numbers in Theorem 1 of this paper has been implemented in a KASH program...