The canonical representation of the Drinfeld curve.

If C$C$ is a smooth projective curve over an algebraically closed field F$\mathbb {F}$ and G$G$ is a group of automorphisms of C$C$, the <italic>canonical representation of</italic> C$C$ is given by the induced F$\mathbb {F}$‐linear action of G$G$ on the vector space H0C,ΩC$H^0\left(C,\Omega _C\right)$ of holomorphic differentials on C$C$. Computing it is still an open problem in general when the cover C→C/G$C \rightarrow C/G$ is wildly ramified. In this paper, we fix a prime power q$q$, we consider the Drinfeld curve, that is, the curve C$C$ given by the equation XYq−XqY−Zq+1=0${XY^q-X^qY-Z^{q+1}=0}$ over F=Fq¯$\mathbb {F}=\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\mathbb {F}_q\hspace{-0.83328pt}}\hspace{0.83328pt}$ together with its standard action by G=SL2Fq${G=SL_2\left(\mathbb {F}_q\right)}$, and decompose H0C,ΩC$H^0\left(C,\Omega _C\right)$ as a direct sum of indecomposable representations of G$G$, thus solving the aforementioned problem in this case. [ABSTRACT FROM AUTHOR]