Endomorphism Rings and Isogenies Classes for Drinfeld Modules of Rank 2 Over Finite Fields

Abstract

Let $\Phi$ be a Drinfeld ${\mathbf{F}}_q[T]$-module of rank 2, over a finite field $L$, a finite extension of $n$ degrees of a finite field with $q$ elements ${\mathbf{F}}_q$. Let $m$ be the extension degrees of $L$ over the field ${\mathbf{F}}_q[T]/P$, $P$ is the $\mathbf{F}$-characteristic of $L$, and $d$ the degree of the polynomial $P$. We will discuss about a many analogies points with elliptic curves. We start by the endomorphism ring of a Drinfeld ${\mathbf{F}}_q[T]$-module of rank 2, End${}_L\Phi$, and we specify the maximality conditions and non maximality conditions as a ${\mathbf{F}}_q[T]$-order in the ring of division End${}_L\Phi {\otimes }_{{\mathbf{F}}_q[T]}$, in the next point we will interested to the characteristic polynomial of a Drinfeld module of rank 2 and used it to calculate the number of isogeny classes for such module, at last we will interested to the Characteristic of Euler-Poincare ${\chi }_{\Phi }$ and we will calculated the cardinal of this ideals.

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