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.