My previous question was closed very badly for asking the broad and deep things, so I now understand the consequences of asking such questions, so I refrain from asking such questions, so this is not a deep question. Please answer it at least.
Question set :
- Why were Tamagawa numbers introduced in mathematics ? , I know the wiki-link which some of users may refer says that " Adeles were introduced by Claude Chavelley .... " and that was not clear, so can anybody give the clear and detailed purpose for introducing the Tamagawa number.
- Next one is " Why are Tamagawa numbers so important in linking the Group theory (computation on groups) to Quadratic theory ? . why does computing Tamagawa numbers consider to give many intuitions and important aspects ? .
And the main question that comes into my mind is
Doubt on formulation
" What role does the Tamagawa numbers play in the theory of elliptic curves ? (Does it comment about the cardinality of $E(\mathbb{Q})$ ? ) " .
To add something my previous post in MO got a fantastic answer by Prof.Kevin Buzzard which is here (I thank Kevin once more for his marvelous post) , but in that post Kevin just mentions that they (i.e. formulators) initially concentrated on computing the Tamagawa-numbers of Elliptic-curves and see what is the analogue of them in that case at-least.
But I think that there must be surely a motive in the minds of Prof.Bryan Birch and Prof.Peter Swinnerton-Dyer because they started with the computation of the Tamagawa number, if we think that the motive was that " Tamagawa number was the invariant, then there are many invariants of Elliptic curves, so probably they might have concentrated on others too, but I think there is some purpose that Tamagawa number serves.
So I am eagerly waiting for another fantastic description explaining both about the background of Tamagawa numbers and also the motive behind the formulators, and why did they choose using the ingredient Tamagawa number in cooking the recipe (B.S.D conjecture) ? .
P.S : The answer will still appear good if some person who know about the background ( I mean the one who talked with the formulators and discussed about the background or the people who have heard of Prof.Bryan Birch talking about the history of conjecture can surely answer this in a beautiful way ) takes initiative. But I doubt whether Prof.Kevin is present here or not, I would be happy is this question someway reaches him, and if answers/comments this, as he has talked with the formulators well. But I am happy and willing that Prof.Matthew Emerton gives another good answer for this too.
Thanks a lot.