$\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 ... (...).
Sur la Structure de A-module de Drinfeld de rang 2