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In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that

There is not merely a local-global principle for curves of genus-$0$, but it has a quantitative formulation ( and also, more generally for linear algebraic groups. The modern formulation is in terms of the " Tamagawa number " ) .

Can any person please help me in understanding the above sentence by expanding it in more clear words, I mean I am looking for an explanation that how can the Modern-Formulation of Tamagawa Number act as a Local-Global Principle.

But I never have any view how can one use the Tamagawa-number as Local-global principle, it seems very interesting for me.

This is the major confusion I have in my mind, I tried writing to many people , but due to scarcity of people working in this area I didn't get an answer.

If anybody helps me I will be much thankful to them.

And I am also looking for beautiful articles on Tamagawa numbers, can anyone provide a reference.

Edit: Can I request Prof.Mathew Emerton to see this question and answer it if he is free.

Thanking you all.

Yours truly,

Iyengar.

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2  
Hi iyengar, you're confusing elliptic curves with their principal homogeneous spaces. The fact that elliptic curves have a rational point by assumption doesn't imply the local-global principle...indeed the non-triviality of the Tate-Shafarevich group for many curves provides counter-examples. –  Cam McLeman Jan 10 '12 at 0:37
    
Sir, thanks a lot for response, but I heard many people ( experts, I just don't want to reveal their names here ) saying that Local-Global Principle always hold for elliptic curves, I too pointed out the same question about the non-triviality of TS group, which is the measure the extent a local point lifts to a global point, later they didn't reply me. So can you please answer this ? , or I need to wait for Prof.Emerton @CamMcLeman –  Iyengar Jan 10 '12 at 15:49
    
Have you looked at the wikipedia page on the Tate-Shafarevich group? There is a good bur brief discussion there. Or in Silverman's Arithmetic of Elliptic Curves? Proposition 6.5, for example, gives explicit curves with non-trivial Tate-Shafarevich groups. –  Cam McLeman Jan 10 '12 at 17:36
    
Yes sir, I have looked both of them, but I have problem in understanding the sentence pointed out by Cassel's @CamMcLeman –  Iyengar Jan 10 '12 at 17:48
    
Me too, as written. I believe you missed a "not" when transcribing it, which might be the source of the confusion. –  Cam McLeman Jan 10 '12 at 18:13

2 Answers 2

This can't be explained in a sentence or two; in fact making sense of this statement takes a whole seminar. For a somewhat down-to-earth approach see Appendix B in Cassels' book on rational quadratic forms available from Dover. In any case one can summarize Siegel's book "Lectures on the analytical theory of quadratic forms" by computing suitable Tamagawa numbers.

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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.

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