I'm quite curious how Birch and Swinnerton-Dyer formed their famous conjecture in the beginning of 1960s. I read some papers of Birch and Swinnerton-Dyer, as well as some papers of Tate and several posts in MO and MATH.SE, which makes the intuition of some factors(e.g. torsion groups, Tamagawa numbers) in the refined BSD conjecture much clearer to me.
Question:
Harold Davenport helped Birch and Swinnerton-Dyer in writing down the explicit formula of $L(E,1)$ by Weierstrass $P$-functions, in which the factors of torsion group and real period naturally emerge. They succeeded in finding out: "after the junk factors had been scrapped off"(i.e., Tamagawa numbers, cardinal of torsion group, real period), the value $L(E,1)$ is some integer equal to the order of Tate-Shafarevich group.
Why did they believe that the integer is the order of some group whose calculation is intangible today?
I read the notes on elliptic curves by B and S-D, and I can hardly find a word about the canonical height and regulator of elliptic curves. I hear that the factor of regulator was established by John Tate around 1965. I am confused how he succeeded in the formation of regulator part in the refined BSD conjecture.
What is the intuition in the formation of the regulator part?
A brief partial answer: you may want to read about Dedekind's formula for the special value of the $\zeta$-function of a number field at $s = 0$. The analogy with BSD (including the role of the regulator) should be fairly clear.