Comptes Rendus
Algebraic Geometry
The A-module structure induced by a Drinfeld A-module of rank 2 over a finite field
Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 305-308.

Let Fq be a finite field and let L/Fq be a finite extension. Let F be the Frobenius of L (F:xx#L) and let (P) be the F[T]-characteristic of F. Let m be the degree of the extension L/Fq[T]/(P). There exists then cFq[T] and μFq such that the characteristic polynomial PF of F is equal to PF(X)=X2cX+μPm. Our main result is an analogue of Deuring's Theorem on elliptic curves: let M=Fq[T](i1)Fq[T](i2), where i1 and i2 are two polynomials of Fq[T] such that i2|i1 and i2|(c2), there exists an ordinary Drinfeld Fq[T]-module Φ of rank 2 over L such that the structure of the finite Fq[T]-module LΦ induced by Φ over L is isomorphic to M.

Soit Fq un corps fini et L/Fq une extension finie. Soit F le Frobenius de L (F:xx#L) et (P) la F[T]-caractéristique de F. Soit m le degré de l'extension L/Fq[T]/(P). Il existe alors cFq[T] et μFq tels que le polynôme caractéristique PF de F soit égal à PF(X)=X2cX+μPm. Notre résultat principal est un parfait analogue du théorème de Deuring pour les courbes elliptiques : soit M=Fq[T](i1)Fq[T](i2), où i1 et i2 sont deux polynômes de Fq[T] tels que i2|i1 et i2|(c2). Il existe alors un Fq[T]-module de Drinfeld Φ ordinaire de rang 2 sur L tel que la structure du Fq[T]-module fini LΦ induite par Φ sur L soit isomorphe à M.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.01.012

Mohamed-Saadbouh Mohamed-Ahmed 1

1 Département de mathématiques, Université du Maine, avenue Olivier-Messiaen, 72085 Le Mans cedex 9, France
@article{CRMATH_2008__346_5-6_305_0,
     author = {Mohamed-Saadbouh Mohamed-Ahmed},
     title = {The {\protect\emph{A}-module} structure induced by a {Drinfeld} {\protect\emph{A}-module} of rank 2 over a finite field},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {305--308},
     publisher = {Elsevier},
     volume = {346},
     number = {5-6},
     year = {2008},
     doi = {10.1016/j.crma.2008.01.012},
     language = {en},
}
TY  - JOUR
AU  - Mohamed-Saadbouh Mohamed-Ahmed
TI  - The A-module structure induced by a Drinfeld A-module of rank 2 over a finite field
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 305
EP  - 308
VL  - 346
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2008.01.012
LA  - en
ID  - CRMATH_2008__346_5-6_305_0
ER  - 
%0 Journal Article
%A Mohamed-Saadbouh Mohamed-Ahmed
%T The A-module structure induced by a Drinfeld A-module of rank 2 over a finite field
%J Comptes Rendus. Mathématique
%D 2008
%P 305-308
%V 346
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2008.01.012
%G en
%F CRMATH_2008__346_5-6_305_0
Mohamed-Saadbouh Mohamed-Ahmed. The A-module structure induced by a Drinfeld A-module of rank 2 over a finite field. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 305-308. doi : 10.1016/j.crma.2008.01.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.012/

[1] B. Angles One some subring of Ore polynomials connected with finite Drinfeld modules, J. Algebra, Volume 181 (1996) no. 2, pp. 507-522

[2] V.G. Drinfeld Elliptique modules, Math. USSR Sb., Volume 94 (1974) no. 136, pp. 594-627 (656)

[3] M. Deuring Die Typen der Multiplikatorenringe Ellipticher Funktionenkorper, Abh. Math. Sem. Univ. Hamburg, Volume 14 (1941), pp. 197-272

[4] D. Goss Basic Structures of Function Field Arithmetic, A Series of Modern Surveys in Mathematics, vol. 35, Springer, 1996

[5] I. Reiner Maximal Orders, Academic Press, 1975

[6] R. Shoof Nonsingular plane cubic curves over finite fields, J. Combinatory Theory Ser. A, Volume 46 (1987), pp. 183-211

[7] M.A. Tsfasman; S.G. Vladut Algebraic-Geometric Codes, Math. Appl., Kluwer, Dordrecht, 1991

[8] J.-K. Yu Isogenis of Drinfeld modules over finite fields, J. Number Theory, Volume 54 (1995) no. 1, pp. 161-171

Cited by Sources:

Comments - Policy