Sur la Structure de A-module de Drinfeld de rang 2

Abstract

$\Phi $ be a Drinfeld $\mathbf{F}\_{q}[T]$-module of rank 2, over a finite field $L$. Let $P\_{\Phi}(X)=$ $X^{2}-cX+\mu P^{m}$ ($c$ an element of $\mathbf{F}\_{q}[T],$ $\mu $ be a non-vanishing element of $% \mathbf{F}\_{q}$, $m$ the degree of the extension $L$ over the field $% \mathbf{F}\_{q}[T]/P,$ and $P$ the $\mathbf{F}\_{q}[T]$-characteristic of $% L $ and $d $ the degree of the polynomial $P$) the characteristic polynomial of the Frobenius $F$ of $L$. We will be interested in the structure of finite $\mathbf{F}\_{q}[T]$-module $L^{\Phi}$ induced by $\Phi $ over $L$. Our main result is analogue to that of Deuring (see \cite{Deuring}) for elliptic curves : Let $M=\frac{\mathbf{F}\_{q}[T]}{I\_{1}}\oplus \frac{\mathbf{F}\_{q}[T]}{% I\_{2}}$, where $I\_{1}=(i\_{1})$,$ I\_{2}=(i\_{2})$ ($i\_{1}$, $i\_{2}$ being two polynomials of $\mathbf{F}\_{q}[T]$) such that : $i\_{2}\mid (c-2)$. Then there exists an ordinary Drinfeld $\mathbf{F}\_{q}[T]$-module $\Phi $ over $L$ of rank 2 such that : $L^{\Phi}$ $\simeq M$. To cite this article: Mohamed-Saadbouh Mohamed-Ahmed, C. R. Acad. Sci. Paris, Ser. I ... (...).