{"title":"二次残差模式、代数曲线和 K3 曲面","authors":"Valentina Kiritchenko , Michael Tsfasman , Serge Vlăduţ , Ilya Zakharevich","doi":"10.1016/j.ffa.2024.102517","DOIUrl":null,"url":null,"abstract":"<div><div>Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive <em>ℓ</em>-tuples of quadratic residues, studying corresponding algebraic curves and their Jacobians, which happen to be products of Jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102517"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Quadratic residue patterns, algebraic curves and a K3 surface\",\"authors\":\"Valentina Kiritchenko , Michael Tsfasman , Serge Vlăduţ , Ilya Zakharevich\",\"doi\":\"10.1016/j.ffa.2024.102517\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive <em>ℓ</em>-tuples of quadratic residues, studying corresponding algebraic curves and their Jacobians, which happen to be products of Jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.</div></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":\"101 \",\"pages\":\"Article 102517\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-10-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Finite Fields and Their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1071579724001564\",\"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/S1071579724001564","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Quadratic residue patterns, algebraic curves and a K3 surface
Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive ℓ-tuples of quadratic residues, studying corresponding algebraic curves and their Jacobians, which happen to be products of Jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.
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.