It had been much more fun to read short proofs in this page itself. Figuring out Russian is a headache if you are a non-Russian. It is hard enough even if you are a Russophile, but not Russophile enough to learn the language. Of course, all this is unless you were at Princeton, and you had to pass an exam in that course!

Shimura is the standard reference. Also he is the prover of all this stuff and creator of many streams of thought in the subject. Unfortunately his book is unreadable. See my answer below and the comments along with it.

@Ben. I have added the word "also" to the title. This should pre-empt the particular objection of yours.

I make apologies in advance for the polemics. But every person has a personal taste, and this is mine. There are people who don't like Shakespeare, though he is very reputed. If you have objections to me, please ignore me as a philistine like one of those guys.

I saw you complaining about Shimura and that is why I pointed this one out. Diamond and Shurman is "too soft" in a sense. But Shimura uses the hopelessly old language for algebraic geometry(Weil's foundations), and is unreadable. Hida has rewritten all that Grothendieck-style in his "Geoemtric modular forms" book; but if anything that is even more of a mess to figure out and is completely unreadable. The Antwerp volumes are by far the best I know of, though a bit old.

But the Tate algorithm is for just one prime. The PARI algorithm is perhaps a more heuristic one, considering all primes at once.

This is cool. The non-trivial analysis statement used is the intermediate value theorem. And there is an explicit expression for square roots, using the completeness of $\mathbb{R}$. I really like this proof.

The second one is quite interesting, and new for me. Thanks. The first one is not exactly the same as in Ahlfors book; but some arguments are common to both.

The moderator Anton revived that question after Anweshi's request.

A comment: It is mentioned in pages about Drinfel'd that he proved the functional field Langlands. Lafforgue is perhaps more general?

If $\mathbb{R}$ is complete, so are the $\mathbb{Q}_p$ for each prime $p$.

@Ben. "a coarse moduli space is fine if and only if a fine moduli space exists" is annoying obvious to me since I have looked in GIT. However I am slippery with stuff like stacks. I am not experienced at that level. I am trying to figure such ideas out, and after reading the nLab page(but not any textbook on stacks or research papers), I am posing a question which seemed plausible to me. A helpful answer will enlighten me more on moduli problems and stacks. I hope my level is clear now.

A short intro could be there in J.-P. Serre, Local Fields. Also one could mention the applications to Fontaine theory.

Anweshi deleted it in when he feared that it will be closed. Anweshi was harassed too much and even got a ban for asking questions.

This may be a stupid question. But is there any other online math discussion forum where this type of question will be answered? If so, please give me the URL and I will be most grateful. I had been looking for one and was most unsuccessful. In a blog only the people who blog can post, and the online math discussions forums I could find were all not advanced enough.