AI 中文总结
研究给定椭圆曲线及正定二元二次型，证明在相关二次型表示集合中存在无穷多\(d\)，使扭曲椭圆曲线\(dy^2 = f(x)\)解析秩为1。
AI 中文摘要
给定一条具有魏尔斯特拉斯方程\(y^2 = f(x)\)的椭圆曲线，以及一个正定二元二次型\(Q(u, v)\)。我们证明，在\(Q\)的亏格中由二次型所表示的集合里，存在无穷多个\(d\)，使得扭曲椭圆曲线\(dy^2 = f(x)\)具有解析秩为1。
英文摘要
Given an elliptic curve with Weierstrass equation $y^2=f(x)$, and a positive definite binary quadratic form $Q(u, v)$. We show that there are infinitely many $d$ in the set represented by the quadratic forms in the genus of $Q$ such that the twisted elliptic curve $dy^2=f(x)$ has analytic rank one.
Comments13 pages
