The canonical representation of the Drinfeld curve

Abstract

If $C$ is a smooth projective curve over an algebraically closed field $F$ and $G$ is a group of automorphisms of $C$, the canonical representation of $C$ is given by the induced $F$-linear action of $G$ on the vector space $H^0\left(C,,,{\Omega }_C\right)$ of holomorphic differentials on $C$. Computing it is still an open problem in general when the cover $C\to C/G$ is wildly ramified. In this paper, we fix a prime power $q$, we consider the Drinfeld curve, that is, the curve $C$ given by the equation $X{Y}^q-X^{q}Y-Z^{q+1}=0$ over $F=\overline{F_q}$ together with its standard action by $G=S{L}_2\left(F_q\right)$, and decompose $H^0\left(C,,,{\Omega }_C\right)$ as a direct sum of indecomposable representations of $G$, thus solving the aforementioned problem in this case.