The canonical representation of the Drinfeld curve

Pub Date : 2024-09-02 DOI:10.1002/mana.202200402
Lucas Laurent, Bernhard Köck
{"title":"The canonical representation of the Drinfeld curve","authors":"Lucas Laurent,&nbsp;Bernhard Köck","doi":"10.1002/mana.202200402","DOIUrl":null,"url":null,"abstract":"<p>If <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math> is a smooth projective curve over an algebraically closed field <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathbb {F}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> is a group of automorphisms of <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math>, the <i>canonical representation of</i> <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math> is given by the induced <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathbb {F}$</annotation>\n </semantics></math>-linear action of <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> on the vector space <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>H</mi>\n <mn>0</mn>\n </msup>\n <mfenced>\n <mi>C</mi>\n <mo>,</mo>\n <msub>\n <mi>Ω</mi>\n <mi>C</mi>\n </msub>\n </mfenced>\n </mrow>\n <annotation>$H^0\\left(C,\\Omega _C\\right)$</annotation>\n </semantics></math> of holomorphic differentials on <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math>. Computing it is still an open problem in general when the cover <span></span><math>\n <semantics>\n <mrow>\n <mi>C</mi>\n <mo>→</mo>\n <mi>C</mi>\n <mo>/</mo>\n <mi>G</mi>\n </mrow>\n <annotation>$C \\rightarrow C/G$</annotation>\n </semantics></math> is wildly ramified. In this paper, we fix a prime power <span></span><math>\n <semantics>\n <mi>q</mi>\n <annotation>$q$</annotation>\n </semantics></math>, we consider the Drinfeld curve, that is, the curve <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math> given by the equation <span></span><math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <msup>\n <mi>Y</mi>\n <mi>q</mi>\n </msup>\n <mo>−</mo>\n <msup>\n <mi>X</mi>\n <mi>q</mi>\n </msup>\n <mi>Y</mi>\n <mo>−</mo>\n <msup>\n <mi>Z</mi>\n <mrow>\n <mi>q</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation>${XY^q-X^qY-Z^{q+1}=0}$</annotation>\n </semantics></math> over <span></span><math>\n <semantics>\n <mrow>\n <mi>F</mi>\n <mo>=</mo>\n <mspace></mspace>\n <mover>\n <msub>\n <mi>F</mi>\n <mi>q</mi>\n </msub>\n <mo>¯</mo>\n </mover>\n <mspace></mspace>\n </mrow>\n <annotation>$\\mathbb {F}=\\hspace{0.83328pt}\\overline{\\hspace{-0.83328pt}\\mathbb {F}_q\\hspace{-0.83328pt}}\\hspace{0.83328pt}$</annotation>\n </semantics></math> together with its standard action by <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>=</mo>\n <mi>S</mi>\n <msub>\n <mi>L</mi>\n <mn>2</mn>\n </msub>\n <mfenced>\n <msub>\n <mi>F</mi>\n <mi>q</mi>\n </msub>\n </mfenced>\n </mrow>\n <annotation>${G=SL_2\\left(\\mathbb {F}_q\\right)}$</annotation>\n </semantics></math>, and decompose <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>H</mi>\n <mn>0</mn>\n </msup>\n <mfenced>\n <mi>C</mi>\n <mo>,</mo>\n <msub>\n <mi>Ω</mi>\n <mi>C</mi>\n </msub>\n </mfenced>\n </mrow>\n <annotation>$H^0\\left(C,\\Omega _C\\right)$</annotation>\n </semantics></math> as a direct sum of indecomposable representations of <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math>, thus solving the aforementioned problem in this case.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202200402","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202200402","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

If C $C$ is a smooth projective curve over an algebraically closed field F $\mathbb {F}$ and G $G$ is a group of automorphisms of C $C$ , the canonical representation of C $C$ is given by the induced F $\mathbb {F}$ -linear action of G $G$ on the vector space H 0 C , Ω C $H^0\left(C,\Omega _C\right)$ of holomorphic differentials on C $C$ . Computing it is still an open problem in general when the cover C C / G $C \rightarrow C/G$ is wildly ramified. In this paper, we fix a prime power q $q$ , we consider the Drinfeld curve, that is, the curve C $C$ given by the equation  X Y q X q Y Z q + 1 = 0 ${XY^q-X^qY-Z^{q+1}=0}$ over F = F q ¯ $\mathbb {F}=\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\mathbb {F}_q\hspace{-0.83328pt}}\hspace{0.83328pt}$ together with its standard action by G = S L 2 F q ${G=SL_2\left(\mathbb {F}_q\right)}$ , and decompose H 0 C , Ω C $H^0\left(C,\Omega _C\right)$ as a direct sum of indecomposable representations of  G $G$ , thus solving the aforementioned problem in this case.

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德林菲尔德曲线的典型表示
如果 是一条代数闭域上的光滑投影曲线,并且 是 的自动形群 ,那么 的典型表示是由 的在全形微分向量空间上的诱导线性作用给出的。在一般情况下,当覆盖有大量斜边时,计算它仍是一个未决问题。在本文中,我们固定一个质幂 ,考虑德林费尔德曲线,即由方程 over 及其标准作用给出的曲线,并将其分解为Ⅳ的不可分解表示的直接和,从而解决了这种情况下的上述问题。
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