论文标题
排名至少$ 2 $的椭圆曲线家庭
On a family of elliptic curves of rank at least $2$
论文作者
论文摘要
令$ c_ {m}：y^{2} = x^{3} - m^{2} x +p^{2} q^{2} $是$ \ mathbb {q} $上的椭圆曲线的家族，其中$ m $是$ m $，$ m $是一个正integer and q $ p，q $ $ prime odd prime。我们研究扭转部分和$ C_M（\ Mathbb {Q}）$的排名。 More specifically, we prove that the torsion subgroup of $C_{m}(\mathbb{Q})$ is trivial and the $\mathbb{Q}$-rank of this family is at least $2$, whenever $m \not \equiv 0 \pmod 4$ and $m \equiv 2 \pmod {64}$ with neither $p$ nor $q$ divides $ m $。
Let $C_{m} : y^{2} = x^{3} - m^{2}x +p^{2}q^{2}$ be a family of elliptic curves over $\mathbb{Q}$, where $m$ is a positive integer and $p, q$ are distinct odd primes. We study the torsion part and the rank of $C_m(\mathbb{Q})$. More specifically, we prove that the torsion subgroup of $C_{m}(\mathbb{Q})$ is trivial and the $\mathbb{Q}$-rank of this family is at least $2$, whenever $m \not \equiv 0 \pmod 4$ and $m \equiv 2 \pmod {64}$ with neither $p$ nor $q$ divides $m$.
