Mordell-Weil groups and the Galois module structure of rings of integers. (English) Zbl 0631.14033

Taylor, M. J.

Ill. J. Math. 32, No. 3, 428-452 (1988).

This is a foundational article for the study of the rings of integers of fields obtained by the division of points on abelian varieties with complex multiplication. It is shown that the arising Galois module structures possess a natural group law, and that they are self-dual, with bounded order in the relevant Grothendieck group. Moreover there is an exciting, new conjectural link made between the L-function of the abelian variety and the triviality of the Galois module structure of such rings of integers.
Particular consideration is then given to the case of elliptic curves: firstly it is shown that, under certain conditions, the above mentioned conjecture is satisfied; secondly it is shown that, in a qualitative sense, trivial Galois module structure is relatively rare.

MSC:

14K22	Complex multiplication and abelian varieties
14G05	Rational points
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
14H45	Special algebraic curves and curves of low genus
14H52	Elliptic curves
11R32	Galois theory

Keywords:

Mordell-Weil groups; division of points on abelian varieties with complex multiplication; L-function; triviality of the Galois module structure; elliptic curves