I am exactly copying the answer given by Mr.Charles Matthews given at MO. So I request neither to up-vote this answer or give me bounty. ( As the credit goes to Charles Matthews )
Answer By Charles Matthews :
The Hasse principle works for quadratic forms. As soon as you consider cubic forms, it doesn't work in general. Mathematics tends in these situations to start with a counterexample. If you are very lucky, as here, you get some sort of theory of counterexamples which may be more "serious" as mathematics than the initial questions. This is what happened in algebraic number theory, where uniqueness of factorisation in number fields into prime elements fails, but into prime ideals is successful as a theory.
So what you are doing, roughly speaking, is trying to work the process backwards, to the heuristic before the current theory. With ideal theory this is fairly clearly the wrong idea for learning the subject: no one can learn algebraic number theory without ideal theory or an equivalent theory. The failure of the Hasse principle for cubic forms should be considered a deeper theory, certainly: there is a theory when the number of variables is large; there are attempts to reduce the number of variables required for a Hasse principle; there are uses of the Brauer group to understand the obstruction for cubic surfaces; and there is the theory for cubic curves (i.e. curves of genus 1 if they are non-singular). The case of curves has had the most attention.
I think it is more helpful to accept that "Hasse's viewpoint" was helpful in opening up this area of research, than to query its current role. There is a way "tradition" works in number theory, but it should be regarded as flexible; because we don't yet know enough about the fundamental ways in which the subject develops.
Alternative answer given by Mr. Timo Keller :
The connection between BSD and Tamagawa numbers is established in
S. Bloch, A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture, Invent. Math. Volume 58, Number 1, 65-76
http://www.springerlink.com/content/lhx4674713046053/
Some useful references suggested by Mr.Chandan-Dalawat :
The answer provided by Charles Matthew was a good one giving a brief intuition into the subject, and later on you can refer to the links given by Franz Lemmermeyer, which tells exactly how it can be done. ( But on the other hand, I will advice you to just start mastering the basic number theory books and then go to advanced concepts. If you are really confident enough, that you prove something without learning, then its well and good. All the best, I learnt that I mustn't discourage others, and its your wish whether to follow my advice or not ) .
Thank you.
