Let be a finite field and let be a finite extension. Let F be the Frobenius of L () and let be the -characteristic of F. Let m be the degree of the extension . There exists then and such that the characteristic polynomial of F is equal to . Our main result is an analogue of Deuring's Theorem on elliptic curves: let , where and are two polynomials of such that and , there exists an ordinary Drinfeld -module Φ of rank 2 over L such that the structure of the finite -module induced by Φ over L is isomorphic to M.
Soit un corps fini et une extension finie. Soit F le Frobenius de L () et la -caractéristique de F. Soit m le degré de l'extension . Il existe alors et tels que le polynôme caractéristique de F soit égal à . Notre résultat principal est un parfait analogue du théorème de Deuring pour les courbes elliptiques : soit , où et sont deux polynômes de tels que et . Il existe alors un -module de Drinfeld Φ ordinaire de rang 2 sur L tel que la structure du -module fini induite par Φ sur L soit isomorphe à M.
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Mohamed-Saadbouh Mohamed-Ahmed 1
@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 -
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] One some subring of Ore polynomials connected with finite Drinfeld modules, J. Algebra, Volume 181 (1996) no. 2, pp. 507-522
[2] Elliptique modules, Math. USSR Sb., Volume 94 (1974) no. 136, pp. 594-627 (656)
[3] Die Typen der Multiplikatorenringe Ellipticher Funktionenkorper, Abh. Math. Sem. Univ. Hamburg, Volume 14 (1941), pp. 197-272
[4] Basic Structures of Function Field Arithmetic, A Series of Modern Surveys in Mathematics, vol. 35, Springer, 1996
[5] Maximal Orders, Academic Press, 1975
[6] Nonsingular plane cubic curves over finite fields, J. Combinatory Theory Ser. A, Volume 46 (1987), pp. 183-211
[7] Algebraic-Geometric Codes, Math. Appl., Kluwer, Dordrecht, 1991
[8] Isogenis of Drinfeld modules over finite fields, J. Number Theory, Volume 54 (1995) no. 1, pp. 161-171
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