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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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    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.2012-01-10
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    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 @CamMcLeman2012-01-10
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    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.2012-01-10
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    Yes sir, I have looked both of them, but I have problem in understanding the sentence pointed out by Cassel's @CamMcLeman2012-01-10
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    Me too, as written. I believe you missed a "not" when transcribing it, which might be the source of the confusion.2012-01-10
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    @CamMcLeman : Thanks a lot sir, I have fixed it, I am extremely sorry for the inconvenience , I am in debt with you for what you have done, I once again deeply apologize for my typo, I have fixed it sir.2012-01-11
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    If you mean [J W S Cassels](http://www.gap-system.org/~history/Biographies/Cassels.html), then you have misspelt the name.2012-01-13
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    @iyengar, *Cassels* not *Cassel*.2012-01-13

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