{"title":"具有群代数分解的雅各布品种不是普莱姆品种所能负担得起的","authors":"Benjamín M. Moraga","doi":"10.1016/j.jpaa.2024.107803","DOIUrl":null,"url":null,"abstract":"<div><div>The action of a finite group <em>G</em> on a compact Riemann surface <em>X</em> naturally induces another action of <em>G</em> on its Jacobian variety <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. In many cases, each component of the group algebra decomposition of <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is isogenous to a Prym varieties of an intermediate covering of the Galois covering <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>/</mo><mi>G</mi></math></span>; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is not affordable by Prym varieties; namely, affine groups <span><math><mi>Aff</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> with some exceptions: <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span>, <span><math><mi>q</mi><mo>=</mo><mn>9</mn></math></span>, <em>q</em> a Fermat prime, <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> a Mersenne prime and some particular cases when <span><math><mi>X</mi><mo>/</mo><mi>G</mi></math></span> has genus 0 or 1. In each one of this exceptional cases, we give the group algebra decomposition of <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> by Prym varieties.</div></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Jacobian varieties with group algebra decomposition not affordable by Prym varieties\",\"authors\":\"Benjamín M. Moraga\",\"doi\":\"10.1016/j.jpaa.2024.107803\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The action of a finite group <em>G</em> on a compact Riemann surface <em>X</em> naturally induces another action of <em>G</em> on its Jacobian variety <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. In many cases, each component of the group algebra decomposition of <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is isogenous to a Prym varieties of an intermediate covering of the Galois covering <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>/</mo><mi>G</mi></math></span>; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is not affordable by Prym varieties; namely, affine groups <span><math><mi>Aff</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> with some exceptions: <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span>, <span><math><mi>q</mi><mo>=</mo><mn>9</mn></math></span>, <em>q</em> a Fermat prime, <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> a Mersenne prime and some particular cases when <span><math><mi>X</mi><mo>/</mo><mi>G</mi></math></span> has genus 0 or 1. In each one of this exceptional cases, we give the group algebra decomposition of <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> by Prym varieties.</div></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022404924002007\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022404924002007","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
有限群 G 在紧凑黎曼曲面 X 上的作用自然会引起 G 在其雅各布综 J(X) 上的另一个作用。在许多情况下,J(X) 的群代数分解的每个分量都与伽罗瓦覆盖πG:X→X/G 的中间覆盖的 Prym 变项同源;在这种情况下,我们说群代数分解是由 Prym 变项负担得起的。在这篇文章中,我们提出了一个无穷群族,这些群族作用于黎曼曲面时,J(X) 的群代数分解不能由 Prym varieties 承担;即仿射群 Aff(Fq),但有一些例外情况:q=2,q=9,q 是费马素数,q=2n,2n-1 是梅森素数,以及 X/G 属 0 或 1 的一些特殊情况。在每一种特殊情况下,我们都给出了 J(X) 的普赖姆变项的群代数分解。
Jacobian varieties with group algebra decomposition not affordable by Prym varieties
The action of a finite group G on a compact Riemann surface X naturally induces another action of G on its Jacobian variety . In many cases, each component of the group algebra decomposition of is isogenous to a Prym varieties of an intermediate covering of the Galois covering ; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of is not affordable by Prym varieties; namely, affine groups with some exceptions: , , q a Fermat prime, with a Mersenne prime and some particular cases when has genus 0 or 1. In each one of this exceptional cases, we give the group algebra decomposition of by Prym varieties.