纪录片《费马最后的定理》 第09期 关于模形式（在线收听）

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, subtraction, multiplication, division and modular forms.

加，减，乘，除以及模形式。

Modular forms are functions on the complex plane that are inordinately 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 mere 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 whatsoever to do with the humdrum 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。