The Galois Structure of the Spaces of polydifferentials on the Drinfeld Curve

Abstract

Let $C$ be a smooth projective curve over an algebraically closed field $\mathbb{F}$ equipped with the action of a finite group $G$. When $p=\text{char}(\mathbb{F})$ divides the order of $G$, the long-standing problem of computing the induced representation of $G$ on the space $H^0(C,{\Omega }_C^{\otimes m})$ of globally holomorphic polydifferentials remains unsolved in general. In this paper, we study the case of the group $G={\mathrm{SL}}_2({\mathbb{F}}_q)$ (where $q$ is a power of~$p$) acting on the Drinfeld curve $C$ which is the projective plane curve given by the equation $X{Y}^q-X^{q}Y-Z^{q+1}=0$. When $q=p$, we fully decompose $H^0(C,{\Omega }_C^{\otimes m})$ as a direct sum of indecomposable $\mathbb{F}[G]$-modules. For arbitrary $q$, we give a partial decomposition in terms of an explicit $\mathbb{F}$-basis of $H^0(C,{\Omega }_C^{\otimes m})$. Finally, in the appendix, we compute the $a$-number and $p$-rank of the Drinfeld curve.