Conjecture

There exist elliptic curve groups $$E(\Bbb Q)$$ of arbitrarily large rank.

Martin-McMillen 2000 年有一个 $$r\geq24$$ 的例子:

\begin{equation*}\begin{split}y^2+xy+y&=x^3-120039822036992245303534619191166796374x\\&+ 504224992484910670010801799168082726759443756222911415116\end{split}\end{equation*}

Hasse-Weil $$L$$-function $$L(s, E)$$ 在 $$s=1$$ 处的零点的阶数 $$r_a$$ 称为 $$E$$ 的 analytic rank(解析秩).

Manjul Bhargava, Christopher Skinner, Wei Zhang(张伟) 7 月 7 日在 arXiv 上传的论文 “A majority of elliptic curves over $$Q$$ satisfy the Birch and Swinnerton-Dyer conjecture“, 宣布了取得的进展:

1. $$\Bbb Q$$ 上的椭圆曲线, when ordered by height(同构类以高排序), 至少有 $$66.48\%$$ 满足 BSD conjecture;
2. $$\Bbb Q$$ 上的椭圆曲线, when ordered by height, 至少有 $$66.48\%$$ 有有限 Tate–Shafarevich group;
3. $$\Bbb Q$$ 上的椭圆曲线, when ordered by height, 至少有 $$16.50\%$$ 满足 $$r=r_a=0$$, 至少有 $$20.68\%$$ 满足 $$r=r_a=1$$.

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