{"title":"德林菲尔德曲线的典型表示","authors":"Lucas Laurent, 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":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"4115-4120"},"PeriodicalIF":0.8000,"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":"{\"title\":\"The canonical representation of the Drinfeld curve\",\"authors\":\"Lucas Laurent, 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\":49853,\"journal\":{\"name\":\"Mathematische Nachrichten\",\"volume\":\"297 11\",\"pages\":\"4115-4120\"},\"PeriodicalIF\":0.8000,\"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\":\"Mathematische Nachrichten\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.202200402\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202200402","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
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。
The canonical representation of the Drinfeld curve
If is a smooth projective curve over an algebraically closed field and is a group of automorphisms of , the canonical representation of is given by the induced -linear action of on the vector space of holomorphic differentials on . Computing it is still an open problem in general when the cover is wildly ramified. In this paper, we fix a prime power , we consider the Drinfeld curve, that is, the curve given by the equation over together with its standard action by , and decompose as a direct sum of indecomposable representations of , thus solving the aforementioned problem in this case.
期刊介绍：
Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index
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