The canonical representation of the Drinfeld curve

Abstract

If $C$ is a smooth projective curve over an algebraically closed field $\mathbb{F}$ and $G$ is a subgroup of automorphisms of $C$, then $G$ acts linearly on the $\mathbb{F}$-vector space of holomorphic differentials $H^0(C,{\Omega }_C)$ by pulling back differentials. In other words, $H^0(C,{\Omega }_C)$ is a representation of $G$ over the field $\mathbb{F}$, called $\text{the canonical representation}$ of $C$. Computing its decomposition as a direct sum of indecomposable representations is still an open problem when the ramification of the cover of curves $C⟶C/G$ is wild. In this paper, we compute this decomposition for $C$ the Drinfeld curve $X{Y}^q-X^{q}Y-Z^{q+1}=0$, $\mathbb{F}={\bar{\mathbb{F}}}_q$, and $G=S{L}_2\left({\mathbb{F}}_q\right)$ where $q$ is a prime power.

9 pages, to appear in Mathematische Nachrichten

References in corpus (1)

• Galois structure of the holomorphic differentials of curves