The canonical representation of the Drinfeld curve

TL;DR

This paper determines the decomposition of the canonical representation of the Drinfeld curve under the action of SL_2 over a finite field, addressing a complex problem involving wild ramification.

Contribution

It explicitly computes the canonical representation decomposition for the Drinfeld curve with a specific automorphism group, advancing understanding of wild ramification cases.

Findings

Decomposition of the canonical representation for the Drinfeld curve is explicitly obtained.

Provides new insights into the structure of automorphism actions on algebraic curves.

Addresses a previously open problem in the context of wild ramification.

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 $H^0\big(C,\Omega_C\big)$ by pulling back differentials. In other words, $H^0\big(C,\Omega_C\big)$ is a representation of $G$ over the field $\mathbb{F}$, called $\textit{the canonical representation}$ 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 $C \longrightarrow C/G$ is wild. In this paper, we compute this decomposition for $C$ the Drinfeld curve ${XY^q-X^qY-Z^{q+1}=0}$, $\mathbb{F}=\bar{\mathbb{F}}_q$, and ${G=SL_2\big(\mathbb{F}_q\big)}$ where $q$ is a prime power.