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 , $m$ the degree of the extension $L$ over the field and $P$ the $\mathbf{F}\_q[T]$-characteristic of 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 ... (...).