𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Complex Multiplication Structure of Elliptic Curves

✍ Scribed by H.W. Lenstra; Jr.

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
523 KB
Volume
56
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

Let k be a finite field and let E be an elliptic curve over k. In this paper we describe, for each finite extension l of k, the structure of the group E(l) of points of E over l as a module over the ring R of endomorphisms of E that are defined over k. If the Frobenius endomorphism ? of E over k does not belong to the subring Z of R, then we find that E(l)$RΓ‚R(? n &1), where n is the degree of l over k; and if ? does belong to Z then E(l) is, as an R-module, characterized by E(l) Γ„ E(l )$RΓ‚R(? n &1). The arguments used in the proof of these statements generalize to yield a description of the group of points of an elliptic curve over an algebraically closed field as a module over suitable subrings of the endomorphism ring of the curve. It is shown that straightforward generalizations of the results of this paper to abelian varieties of dimension greater than 1 cannot be expected to exist.

πŸ“œ SIMILAR VOLUMES

Galois Structure of Rings of Integers an
Galois Structure of Rings of Integers and Elliptic Curves without Complex Multiplication
✍ A. Bayad πŸ“‚ Article πŸ“… 1995 πŸ› Elsevier Science 🌐 English βš– 408 KB

In this paper we state the problem of Galois module structure of rings of integers of extensions attached to the elliptic curves without complex multiplication and admitting a rational point of finite order. Our main aim is to give the first results related to this problem. These results are an anal

Elliptic Curves with Complex Multiplicat
Elliptic Curves with Complex Multiplication and a Character Sum
✍ R. Padma; S. Venkataraman πŸ“‚ Article πŸ“… 1996 πŸ› Elsevier Science 🌐 English βš– 336 KB

&163 ). We want to calculate the character sum where ( } Γ‚p) is the Legendre symbol. We show that it is enough to calculate the sum for a finite number of primes and this has been done.

Character Sums Linked To Elliptic-Curves
Character Sums Linked To Elliptic-Curves With Complex Multiplication
✍ A. Joux; F. Morain πŸ“‚ Article πŸ“… 1995 πŸ› Elsevier Science 🌐 English βš– 605 KB

Nous évaluons les sommes de caractères liées aux courbes elliptiques à multiplication complexe par l'anneau des entiers d'un corps quadratique imaginaire de discriminant \(-43,-67\) ou -163 , en simplifiant la méthode due à Rajwade et alii. 1995 Academic Press. Inc.