纪录片《费马最后的定理》 第09期 关于模形式

（单词翻译）

Together, Taniyama and Shimura worked on the complex mathematics of modular functions.

谷山丰和志村五郎一起进行模函数方面复杂数学的研究

I really can't explain what a modular function is in one sentence.

我实在不能用一句话来描述模函数是什么

I can try and give you a few sentences to explain it.

我可以试着以几句话来解释

I really can't put it in one sentence.

确实不能以一句概括之

Oh it's impossible.

哦，不可能的。

There's a saying attributed to Eichler that there are five fundamental operations of arithmetic:

埃其勒曾说过算术中有五种基本运算:

addition, subtraction¹, multiplication², division and modular forms.

加，减，乘，除以及模形式。

Modular forms are functions on the complex plane that are inordinately³ symmetric.

模形式是有着非比寻常对称性的复杂平面的函数

They satisfy so many internal symmetries that their mere⁴ existence seem like accidents, but they do exist.

它们具有如此多的内在对称性，其存在看来像出自意外，但它们确实存在。

This image is merely a shadow of a modular form.

这幅图像仅是一个模形式的映射

To see one properly your TV screen would have to be stretched into something called hyperbolic space.

要看得真切，你的电视屏可能得需要伸展入双曲空间。

Bizarre modular forms seem to have nothing whatsoever⁵ to do with the humdrum⁶ world of elliptic curves.

怪里怪气的模形式看上去和椭圆曲线的单调世界没什么关系。

But what Taniyama and Shimura suggested shocked everyone.

但谷山丰和志村五郎的提法语惊四座。