Tamagawa Number of Elliptic Curves over $\mathbb{Q}$

Question

I am currently reading a paper by De Weger and one theorem in it proves a bound for the Tamagawa number of any elliptic curve defined over $\mathbb{Q}$.

I was wondering if anyone has any good references/texts that provide an exposition on the Tamagawa number of an elliptic curve as I was unable to find one in the Arithmetic of Elliptic Curves.

I know the definition of the Tamagawa number from this reference, but not really much more than that (http://math.uci.edu/~asilverb/connectionstalk.pdf).

EDIT: I am looking for something a little more in depth than the intuition behind the definition. For instance useful recent applications of Tamagawa numbers or what is known about Tamagawa numbers for elliptic curves over $\mathbb{Q}$.

Answer 1

I think I would start with Weil "Adeles and algebraic groups", but if you are looking for something more specifically associated to elliptic curves, maybe this survey of Guido Kings: http://epub.uni-regensburg.de/13613/1/MP6.pdf is a good starting point to see the connection between the Equivariant Tamagawa Number conjecture and the BSD conjecture, and consider the references therein.

And here is anothor one of M.Flach: http://www.math.caltech.edu/papers/baltimore-final.pdf

Answer 2

Look at Intuition behind the Tamagawa numbers : Tate's article in Antwerp IV (Springer Lecture Notes in Mathematics 476), Modular Functions of One Variable IV.