纪录片《费马最后的定理》 第16期 伽罗华表示

（单词翻译）

Andrew was embarking¹ on one of the most complex calculations in history.

安德鲁要进行的是史上最为复杂的计算之一。

For the first two years, he did nothing but immerse himself in the problem, trying to find a strategy which might work.

在头两年，他其它的事什么都没做，只是埋首于难题当中，试图找到有用的策略。

So it was now known that Taniyama-Shimura implied Fermat's last theorem.

现在已经知道谷山-志村猜想暗示了费马最后定理。

What does Taniyama-Shimura say?

谷山-志村猜想是怎么说的?

It says that all elliptic curves should be modular.

它说所有的椭圆曲线都应为模形式

Well this was an old problem, been around for 20 years and lots of people would try to solve it.

这是个老难题了，已有20年左右，许多人尝试解决它。

Now one way of looking at it is that you have all elliptic curves and then you have the modular elliptic curves

而如今看待它的一种方式就是，你有了所有的椭圆曲线，接着你又有了模形式椭圆曲线，

and you want to prove that there are the same number of each.

你所要证明的就是这两者有同样的数量。

Now of course you're talking about infinite sets, so you can't just can't count them per say,

当然了，所谈及的是无限集合，因此当然不能仅靠计数，

but you can divide them into packets and you could try to count each packet and see how things gone

但你可将它们分成一族族的，这样就可尽量来数每个族，看看会如何，

and this proves to be a very attractive idea for about 30 seconds

头30秒这是个相当吸引人的想法

but you can't really get much further than that,

但其实你也只能到那一地步了

and the big question on the subject was how you could possibly count, and in fact, Wiles² introduced the correct technique.

而关于这个的关键问题在于你如何来计数，实际上，怀尔斯引入了正确的方法。

Andrew's trick was to transform the elliptic curves into something called Galois representations which would make counting easier.

安德鲁的方法是将椭圆曲线转化为称为伽罗华表示的形式，这能使计数容易些。

Now it was a question of comparing modular forms with Galois representations, not elliptic curves.

现在是将模形式和伽罗华表示、而不是和椭圆曲线进行比较的问题了。