Sign functions of imaginary quadratic fields and applications

Elliptic units and sign functions

“…In [5] we introduced a certain resolution (L m , d 1 ) of U 0 sgn (m), analogue of Anderson's resolution of the universal ordinary distribution U r of level r ∈ N * , discovered by Anderson in [6], Appendix. In [7], Ouyang uses Anderson's resolution to produce spectral sequences convergent toĤ i (I, U r ), where I = Gal(Q(ζ r )/Q(ζ r + ζ −1 r )) and ζ r is a primitive r-th root of 1.…”

“…If L is any finite abelian extension of k and b is a fractional ideal of k prime to the conductor of L/k then we denote by (b, L/k) the k-automorphism of L associated to b by the Artin map. In [5] we introduced the following new invariants. Let Cl(m) be the ideal class group of k moduo m. Let C be the class of c. Then we have…”

“…The same conclusion is true if χ is the trivial character of G. Thus we may suppose that χ is a non trivial character of Gal(k n /k). But in this case (10) is easily derived from the Kronecker limit formula as stated in [4] formula (5).…”

“…In this paper we propose a new approach of the construction of E m based on the material introduced in [5]. The fundamental idea is the concept of sign function.…”

Unités elliptiques, indice et \mathbf{Z}_p-extensions

“…Signalons que dans le cas des corps de rayon on peut prouver des ré-sultats nettement plus précis. Voir [4], [5] et [6]. Dans le Théorème 3.16 on rend compte du comportement dans les Z p -extensions du quotient de la p-partie de l'indice des unités elliptiques de Rubin, cf.…”