The Galois Structure of the Spaces of polydifferentials on the Drinfeld Curve
Denver-James Logan Marchment, Bernhard Köck
TL;DR
We address the modular representation problem for the action of ${G=SL_2({\bf F}_q)}$ on the space ${H^0(C,\Omega_C^{\otimes m})}$ of holomorphic polydifferentials on the Drinfeld curve ${C}$ in characteristic ${p}$ with ${p\mid |G|}$. The authors construct an explicit basis ${\omega_{ij}}$ for all ${q=p^r}$ and obtain a partial ${\mathbb F}[G]$-decomposition for general ${q}$, culminating in a complete indecomposable decomposition when ${q=p}$ using Green correspondence with the subgroup ${B}$ of upper-triangular matrices. They provide precise multiplicities for the ${\mathbb F}[B]$-decomposition ${\rm Res}^G_B(H^0(C,\Omega_C^{\otimes m}))$ and lift these to the full ${\mathbb F}[G]$-decomposition, giving explicit formulas for the numbers ${n_{a,b}}$ and ${n_i}$ via ramification data and Euler characteristics. The results advance understanding of the Galois structure of polydifferentials on a central Drinfeld curve and have potential implications for modular representation theory in geometric settings. $H^0(C,\Omega_C^{\otimes m})$ here denotes the space of globally holomorphic polydifferentials on the curve ${C}$, with basis elements built from ${\omega_{ij}}$ and organized by degree modulo ${q+1}$.
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 $p =\textrm{char}(\mathbb{F})$ divides the order of $G$, the long-standing problem of computing the induced representation of $G$ on the space $H^0(C,Ω^{\otimes m}_C)$ 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 $XY^q-X^qY-Z^{q+1} = 0$. When $q = p$, we fully decompose $H^0(C,Ω^{\otimes m}_C)$ 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 $H^0(C,Ω^{\otimes m}_C)$.