{"title":"关于 p2 阶子群的 p 特性赫米蒂曲线的伽罗瓦子覆盖率","authors":"Barbara Gatti , Gábor Korchmáros","doi":"10.1016/j.ffa.2024.102450","DOIUrl":null,"url":null,"abstract":"<div><p>A (projective, geometrically irreducible, non-singular) curve <span><math><mi>X</mi></math></span> defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> is <em>maximal</em> if the number <span><math><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> of its <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-rational points attains the Hasse-Weil upper bound, that is <span><math><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>g</mi><mi>q</mi><mo>+</mo><mn>1</mn></math></span> where <span><math><mi>g</mi></math></span> is the genus of <span><math><mi>X</mi></math></span>. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> where <em>p</em> is the characteristic of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. Doing so we also determine the <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-isomorphism classes of such curves and describe their full <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-automorphism groups.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102450"},"PeriodicalIF":1.2000,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000893/pdfft?md5=4d7dc430a08ae7fb7ea1e9c89490e861&pid=1-s2.0-S1071579724000893-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2\",\"authors\":\"Barbara Gatti , Gábor Korchmáros\",\"doi\":\"10.1016/j.ffa.2024.102450\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A (projective, geometrically irreducible, non-singular) curve <span><math><mi>X</mi></math></span> defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> is <em>maximal</em> if the number <span><math><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> of its <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-rational points attains the Hasse-Weil upper bound, that is <span><math><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>g</mi><mi>q</mi><mo>+</mo><mn>1</mn></math></span> where <span><math><mi>g</mi></math></span> is the genus of <span><math><mi>X</mi></math></span>. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> where <em>p</em> is the characteristic of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. Doing so we also determine the <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-isomorphism classes of such curves and describe their full <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-automorphism groups.</p></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":\"98 \",\"pages\":\"Article 102450\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-06-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S1071579724000893/pdfft?md5=4d7dc430a08ae7fb7ea1e9c89490e861&pid=1-s2.0-S1071579724000893-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Finite Fields and Their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1071579724000893\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724000893","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
如果在有限域 Fq2 上定义的(投影的、几何上不可还原的、非奇异的)曲线 X 的 Fq2 有理点数 Nq2 达到哈塞-韦尔(Hasse-Weil)上限,即 Nq2=q2+2gq+1 其中 g 是 X 的属。迄今为止,对于赫尔墨斯曲线的伽罗瓦覆盖的一些曲线,特别是在伽罗瓦群有素数阶的情况下,人们已经临时完成了这一工作。在本文中,我们得到了赫尔墨斯曲线的所有伽罗华盖的明确方程,这些曲线的伽罗华群为 p2 阶,其中 p 是 Fq2 的特征。为此,我们还确定了这些曲线的 Fq2-同构类,并描述了它们的全 Fq2-同构群。
Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2
A (projective, geometrically irreducible, non-singular) curve defined over a finite field is maximal if the number of its -rational points attains the Hasse-Weil upper bound, that is where is the genus of . An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order where p is the characteristic of . Doing so we also determine the -isomorphism classes of such curves and describe their full -automorphism groups.
期刊介绍:
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.