Quadratic residue patterns, algebraic curves and a K3 surface

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-10-10 DOI:10.1016/j.ffa.2024.102517

Valentina Kiritchenko , Michael Tsfasman , Serge Vlăduţ , Ilya Zakharevich

引用次数: 0

Abstract

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 In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove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.

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二次残差模式、代数曲线和 K3 曲面

自 19 世纪以来，人们一直在研究调制素数的二次残差模式。在第一部分中，我们扩展了关于二次残差的连续 ℓ-tuples 数的现有结果，研究了相应的代数曲线及其雅各比，它们恰好是超椭圆曲线雅各比的乘积。在第二部分中，我们陈述了莉迪亚-冈察洛娃（Lydia Goncharova）最后一个未发表的关于正方形的结果，即它们的差也是正方形，用 K3 曲面的代数几何重新表述并证明了这一结果。该定理的核心是 K3 曲面上的点数与 CM 椭圆曲线上的点数之间的意外关系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

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.