zbMATH — the first resource for mathematics

Some remarks on Iwasawa theory for elliptic curves. (Quelques remarques sur la théorie d’Iwasawa des courbes elliptiques.) (French) Zbl 1047.11056
Bennett, M. A. (ed.) et al., Number theory for the millennium III. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21–26, 2000. Natick, MA: A K Peters (ISBN 1-56881-152-7/hbk). 119-147 (2002).
The author gives an interesting survey of Iwasawa theory of elliptic curves over the cyclotomic $${\mathbb Z}_p$$-extension. She discusses in detail the local exponential map in the three cases of ordinary good reduction, multiplicative reduction, and good supersingular reduction. Combining this with the deep results of Kato on the existence of an Euler system attached to an elliptic curve over the cyclotomic $${\mathbb Z}_p$$-extension of $${\mathbb Q}$$, she discusses in detail the cyclotomic main conjectures for elliptic curves. The main thrust of the paper is the study of a canonical map which she calls the ‘exponential map’ or ‘Iwasawa regulator’. This map generalises the earlier construction due to Iwasawa, Coates-Wiles and Coleman. Combined with the reciprocity law established by Benois, Colmez and Kato-Kurihara-Tsuji, it can be applied on elements coming from an Euler system to give the $$p$$-adic $$L$$-function. For the case of elliptic curves that are supersingular at the prime $$p$$, there is a new interpretation of the trivial zero phenomenon.
For the entire collection see [Zbl 1002.00007].

MSC:
 11G05 Elliptic curves over global fields 11R23 Iwasawa theory