I have been trying to understand the Néron-Tate global canonical height of algebraic points on elliptic curves.
Let $K$ be a number field, $E$ an elliptic curve (over $\mathbb{Q}$, say), and $E(K)$ the Mordell-Weil group of $K$-rational points on $E$.
I understand Néron's construction of the "naive" height as a "quasi-quadratic form" on $E(K)$. An averaging procedure then turns this "quasi-quadratic" form into a quadratic form in a canonical way. This is all pretty straightforward.
What I don't understand very well is Tate's decomposition of the global canonical height as a sum of local heights, taken over the normalized absolute values of $K$. It seems like pure magic to me. What I find most disturbing is the absolute lack of analogy between the construction of the local height for Archimedean absolute values, resp. for non-Archimedean ones. The local height for Archimedean absolute values is given by construction which belongs completely to the realm of analysis, as far as I can see - whereas for non-Archimedean absolute values, it is given by an essentially arithmetic function; and yet these all patch up together to give the canonical height... it's very weird and surprising and I'd like to understand.
The only reference which has been helpful has been Lang's Elliptic Curves: Diophantine Analysis. Unfortunately, Lang is not the most generous writer when it comes to details or intuition. I would really appreciate an accessible reference on the topic, or, of course, some explanations. Thank you!