Jul 242014


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

用 \(r\) 表示 \(\Bbb Q\) 上的椭圆曲线 \(E\) 的秩—the rank of the Mordell–Weil group \(E(\Bbb Q)\).

一个悬而未决的著名难题是: \(r\) 是否可以任意大?

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 \(r=r_a=1\).

谁将在 8 月 13 日的 ICM 2014 开幕式上获得 Fields medal?坊间向来不缺传闻. 数论大牛 Manjul Bhargava 无疑是最耀眼的明星.

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