x3 + x2y2 + y3 = k 上的有理点

IF 0.5 3区数学 Q3 MATHEMATICS

International Journal of Number Theory Pub Date : 2024-04-05 DOI:10.1142/s1793042124500878

Xiaoan Lang, Jeremy Rouse

引用次数: 0

摘要

我们研究的问题是，在给定整数 k 的情况下，确定 Ck:x3z+x2y2+y3z=kz4 的有理解。对于 k≠0，曲线 Ck 的属数为 3，其 Jacobian 与三条椭圆曲线 E1,k、E2,k、E3,k 的乘积同源。在假设其中一条椭圆曲线的秩为零的情况下，我们明确地确定了 Ck 上的有理点。我们讨论了将我们的结果扩展到处理所有 k∈ℚ 所涉及的挑战。

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Rational points on x3 + x2y2 + y3 = k

We study the problem of determining, given an integer $k$ , the rational solutions to $C_{k} : x^{3} z + x^{2} y^{2} + y^{3} z = k z^{4}$ . For $k \neq 0$ , the curve $C_{k}$ has genus $3$ and its Jacobian is isogenous to the product of three elliptic curves $E_{1, k}$ , $E_{2, k}$ , $E_{3, k}$ . We explicitly determine the rational points on $C_{k}$ under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all $k \in ℚ$ .

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来源期刊

International Journal of Number Theory 数学-数学

CiteScore

1.10

自引率

14.30%

发文量

审稿时长

4-8 weeks

期刊介绍： This journal publishes original research papers and review articles on all areas of Number Theory, including elementary number theory, analytic number theory, algebraic number theory, arithmetic algebraic geometry, geometry of numbers, diophantine equations, diophantine approximation, transcendental number theory, probabilistic number theory, modular forms, multiplicative number theory, additive number theory, partitions, and computational number theory.