陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Distinguished Lecture Series II: Shou-wu Zhang, “Gross-Zagier formula and Birch and Swinne」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

On Wednesday, Shou-wu Zhang continued his lecture series. Whereas the first lecture was a general overview of the rational points on curves problem, the second talk focused entirely on the genus 1 case – i.e. the problem of finding rational points on elliptic curves. This is already a very deep and important problem in number theory – for instance, this theory is decisive in Wiles’ proof of Fermat’s last theorem. It was also somewhat more technical than the previous talk, and

NB: the talk here seems to be loosely based in part on Shou-wu’s “Current developments in Mathematics” article from 2001.

已知结果和反例

The problem of finding rational (or more generally, algebraic) points on a curve is clearly an algebraic problem, but a remarkable feature of the subject is that it can be profitable (in the right circumstances) to seek such points via using transcendental functions, which at first glance are much more closely related to analysis and geometry than to algebra. Shou-wu gave the analogy of the unit circle as sort of a simplified toy model of an elliptic curve. One way to constru

The sine function also satisfies a differential equation, which almost defines this function uniquely:

证明或构造的主线

Finally, of course, we all know that the sine function has a very geometric interpretation.

Now for constructing rational points on the unit circle, such as , it is not so profitable to use these transcendental functions, and it is better to use the purely algebraic formula to generate the points. It seems that when we move to elliptic curves, something similar happens: the analogue of the transcendental sine function is the Weierstrass elliptic function , while the analogue of the purely algebraic formula is more subtle (and becomes transcendental also); once again

阅读时建议盯住的点

In the previous lecture , Shou-wu defined an elliptic curve E as a genus 1 curve with a marked point P (which will eventually become the group identity element). It is traditional to move this point P to the (vertical) point at infinity and view the curve affinely, in which case the curve can be placed via an affine transformation in the normal form

(ignoring some minor difficulties arising from characteristic 2 or 3, and assuming that the discriminant is non-zero). As before, we can view this curve over any field k, for instance giving a Riemann surface , the rational points , the finite field points , etc. (There are some minor issues regarding whether one should view E affinely or projectively here; I don’t fully understand this issue and will therefore ignore it.)

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源