Intuition behind the Tamagawa numbers

Question

i have read many books concerning the definition of tamagawa numbers ,but none of the books explained an intuition behind the concept , i mean what could be the intuitive definition of tamagawa number

i am expecting some other explanation ,other than the ones present in the textbooks,

i wanted to know the reason why the "prof.Peter swinnerton dyer " used the tamagawa number as a central part in conjecturing his work about elliptic curves, what is the reason behind that ???

what do we get when we calculate the tamagawa numbers of the elliptic curve ,i mean i want a logical imaginative definition of it,

hope you understood,

Answer 1

The Euler factors in the $L$-series of an elliptic curve at non-singular primes can be defined as integrals over the $p$-adic points of $E$. When one does the analogous integral over $E(\mathbb{Q}_p)$ for singular primes, then one gets the number of components, which is $\#E(\mathbb{Q}_p)/E_0(\mathbb{Q}_p)$, multiplied by the integral over the identity component $E_0(\mathbb{Q}_p)$. The integral over the identity component gives the Euler factor at the singular prime, i.e., one of $1$, $(1-p^{-s})^{-1}$, or $(1+p^{-s})^{-1}$. So there's this extra factor given by the number of components. Of course, the number of components is the Tamagawa number in this setting. This is all explained in detail in Tate's article in Antwerp IV (Springer Lecture Notes in Mathematics 476), which is where I learned about it. I expect this will be easier to read than the Bloch-Kato article.

BTW, in the original formulation of Birch and Swinnerton-Dyer, they didn't know where the extra factors came from, so they just said that they were small integer "fudge factors". I believe it was Tate who indicated why they should be Tamagawa numbers.