# The Néron-Tate canonical height on elliptic curves

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!

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Perhaps Chapter VI of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves" might provide some of the explanation that you're looking for. It appears to at least give some explanation regarding why local heights for non-Archimedean absolute values are constructed this way.

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Thank you very much Michael for the reference, and welcome here! – Bruno Joyal Jul 31 '11 at 19:59
Thanks for the welcome Bruno and glad to be of help to you! – Michael Banaszek Aug 1 '11 at 18:27

The local heights at nonarchimedean places are obtained as follows: you spread out the elliptic curve $E$ over the ring of integers $\mathcal O_{K_v}$, you spread out the two points into curves in this two-dimensional scheme, and then you compute their intersection.

The local height at an archimedean place is obtained by taking the complex points of the elliptic curve, and using a Green's function. However, you can fill out the elliptic curve to a solid torus (this is an analogue of spreading out over $\mathcal O_{K_v}$) and then realize your degree zero divisors (whose local height pairing you are going to compute) as boundaries of $1$-cycles in this solid torus. The Green's function should then have a geometric interpretation in terms of these $1$-cycles. (Sorry not to be more precise here; hopefully someone else can add more details. You can also look at Manin's paper Three dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry, where I think he proves a formula of the type I am suggesting. The rough idea, though, is that we can "spread out" at the infinite primes too, by passing from a torus to a solid torus, and replacing our points by $1$-cycles.)

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Thank you Matt, this seems very interesting. I will take a look! – Bruno Joyal Jul 31 '11 at 19:59
I have accepted Michael's answer as the reference he cited was exactly what I needed. I hope you don't mind; your answer was very much appreciated. – Bruno Joyal Aug 1 '11 at 18:24