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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 :

  1. 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.
  2. 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.

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I am counting 8 explicit questions in your text. –  Phira Nov 26 '11 at 17:59
    
But Even though they are explicit, some are not questions anymore, but they are alternate perspectives to help the person answering (just asked to understand my requirement) @Phira –  Iyengar Nov 26 '11 at 18:02
    
Possible duplicates (by the same user) at: mathoverflow.net/questions/71044/… and mathoverflow.net/questions/66561/… –  Dimitrije Kostic Nov 26 '11 at 18:24
    
Why "resurrect" the question when you accepted correct answers for both of the above questions? –  Dimitrije Kostic Nov 26 '11 at 18:25
    
@DimitrijeKostic : Even though your both links refer to the same words "tamagawa numbers", but I am completely sure that they are not the same as this question –  Iyengar Nov 27 '11 at 3:33
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1 Answer

up vote 13 down vote accepted

There are classical formulas ("genus formulas") in the theory of quadratic forms due to Siegel. Tamagawa saw how to reformulate these formulas in terms of a simple formula for the volume of the quotient $G(\mathbb Q)\setminus G(\mathbb A)$, where $G$ is the orthogonal group attached to the quadratic form in question. (Part of Tamagawa's observation was that there is a canonical measure --- Tamagawa measure --- which one can use to compute this volume; this is something of a surprise, since one normally expects Haar-type measures to be defined only up to a scalar. It reflects the special adelic nature of the situation.)

Weil further pursued Tamagawa's ideas, and realized that a very general statement should be true; for a simply connected, semisimple linear algebraic group $G$ over $\mathbb Q$, the Tamagawa number of $G$ (which is the standard terminology for the volume of $G(\mathbb Q)\setminus G(\mathbb A)$ with respect to Tamagawa measure) should be equal to $1$.

When one actually computes the Tamagawa number of $G$, certain Euler products appear.

The theory of Tamagawa numbers, and Weil's general conjecture about Tamagawa number $1$, were very much in the air in the 1950s, and I think that Birch and Swinnerton-Dyer were curious to see what happened if one took $G$ to be an elliptic curve, rather than a linear group. The Euler product that appeared in computing the Tamagawa number heuristically seems to be related to the value of the $L$-series at $1$. The rest of the story is recounted nicely in Kevin's answer, I think.

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Sir again a fantastic answer, but can I hear something about the Ken Ono's attempt of proving something called "Tamagawa Number of Tori", I think that he wrote the formula analogously for tori, Am I right sir ? , but on the other hand I understood that Birch and Swinnerton-Dyer simply were computing some random calculations, so in their random search they simply tried to write an analogue for Tamagawa number of an elliptic curve, But that thing is attached to the $L$-function by the suggestions of other persons and luckily that turned out to be a parameter .....Contd @Matt E –  Iyengar Nov 27 '11 at 3:43
    
Contd...: That heuristic thing turned out to be a measure of cardinality of $E(\mathbb{Q})$, but what the thing I understood is correct or wrong sir ? , but do you mean that all these things happened luckily, like I know that major inventions in number theory happen due to random trials and pattern search, but is there any Intention behind the formulators that Tamagawa number should result in some sort of measure on Cardinality of $E(\mathbb{Q})$ ? @Matt E –  Iyengar Nov 27 '11 at 3:46
    
Sir I would be still happy if you suggest me any references of Tamagawa numbers, and genus formula etc... ,in a full-fledged form, thank you sir –  Iyengar Nov 27 '11 at 5:03
    
@iyengar: Dear Iyengar, I think that you are confusing Ken Ono and his father Takashi Ono. Yes, T. Ono worked on Tamagawa numbers of tori; these are linear algebraic groups, but not semisimple. Again, studying their Tamagawa numbers leads to $L$-values arising from Euler products, class number formulas, and related things. Whether B and S-D initially expected to see a count of the solutions to $E(\mathbb Q)$ I don't know. It's been a long time since I read (at least one of) their original papers. Regards, –  Matt E Nov 27 '11 at 12:52
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Dear Iyengar, Regarding genus formulas, etc., type "Siegel--Weil formula" into google. Regards, –  Matt E Nov 27 '11 at 12:52
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