Thurston的经典名著。

Mumford 的红宝书，包含他与格罗滕迪克的通信。

# 从模函数到单值化定理 Ⅴ

Prologue:

…It is true that Mr. Fourier had the opinion that the principal purpose of mathematics was the benefit of the society and the explanation of phenomena of nature; but a philosopher like he should know that the sole purpose of science is the honor of the human mind, and under this title, a question about numbers is as valuable as a question about the system of the world…

——C. G. Jacobi, Letter to Legendre

$$f_{i}(z+1)=f_{i}(z)，f_{i}(z+\tau)=e^{-2k\pi iz}f_{i}(z)$$

$S$ 当然不能嵌入 $\mathbb{C}P^{1}$。取 $k=3$，利用上述的 $f_{i}$ 可完成 $S$ 到 $\mathbb{C}P^{2}$ 的嵌入。我们不再讨论技术性的细节，而是指出类似的想法可以推广到高维。高维复环面可以嵌入射影空间当且仅当其周期矩阵满足Frobenius关系。

Weierstrass提出了另一种构造椭圆函数的方法，即利用Weierstrass $\mathfrak{P}$函数。这方法简洁明了，被大多数现代课本采用。然而值得指出的是，theta函数处在数论、自守形式、函数论和数学物理的交叉点上，研究其性质有极高的附加价值。在第7章中有一个重要的例子：尖点形式$\Delta$。

Jacobi noted, as mathematics’ most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum.

These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity between the east coast of America and the west coast of Africa in geology.

The emotional significance of such discoveries for teaching is difficult to overestimate. It is they who teach us to search and find such wonderful phenomena of harmony of the Universe.

$$\pi(bq) > \pi(aq)$$

$$\frac{\pi(bq)}{\pi(aq)}\sim\frac{b\ln(aq)}{a\ln(bq)} =\frac{b(\ln q+\ln a)}{a(\ln q+\ln b)}\sim\frac ba>1.$$

$$\pi(p_n)=n\sim\frac{p_n}{\ln p_n},$$

$$\ln p_n\sim\ln\frac{p_n}{\ln p_n}\sim\ln n.$$

$$\lim_{k\to\infty}\frac{p_{n_k}}{p_{m_k}} = \lim_{k\to\infty}\frac{n_k\ln n_k}{m_k\ln m_k}=a$$