arXiv subjects

Denver-James Logan Marchment

Publications and source records attributed to Denver-James Logan Marchment.

2 recordsLinked to original sources

arXiv2026

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

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 $H^0(C,Ω^{\otimes m}_C)$. Finally, in the appendix, we compute the $a$-number and $p$-rank of the Drinfeld curve.

math.AG↗

arXiv2025

The Green correspondence for SL(2,p)

Let ${p > 2}$ be an odd prime and ${G = SL_2(\mathbb{F}_p)}$. Denote the subgroup of upper triangular matrices as $B$. Finally, let ${\mathbb{F}}$ be an algebraically closed field of characteristic ${p}$. The Green correspondence gives a bijection between the non-projective indecomposable ${\mathbb{F}[G]}$ modules and non-projective indecomposable ${\mathbb{F}[B]}$ modules, realised by restriction and induction. In this paper, we start by recalling a suitable description of the non-projective indecomposable modules for these group algebras. Next, we explicitly describe the Green correspondence bijection by pinpointing the modules' position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of these descriptions: formulae for lifting the ${\mathbb{F}[B]}$ module decomposition of an ${\mathbb{F}[G]}$ module, and a complete description of ${\text{Ind}_B^G}$ and ${\text{Res}^G_B}$.

math.RT↗