Height pairing and the Néron model of an elliptic curve

Question

I have a question on Joseph Silverman's book ``Advanced topics in the arithmetic of elliptic curves’’ (1999 printing). I asked him; he answered that he doesn't know off-hand and suggested that I put it as a question on MathOverflow, which I am doing now.

On p. 251, he defines a subgroup $E(K)_0$ of the group of points of an elliptic curve $E$ over the function field $K$ of a curve $C/k$: the Néron-Tate height pairing is integral when restricted to this subgroup. In Rem. 9.4.1, he says that it will be studied in detail in the next chapter, referring to (IV.6.12), (IV.9.1), (IV.9.2) and exercise 4.25. I hope that $E(K)_0$ is $\mathcal{E}^0(C)$, where $\mathcal{E}$ is the Néron model of $E$ and $\mathcal{E}^0$ is its identity component. But

IV.6.12 does not exist; IV.9.2 mentions a subgroup $E_0(K)$ which is indeed equal to $\mathcal{E}^0(R)$, but the situation is local ($R$ is a dvr) and I cannot find something comparing it to $E(K)_0$ (the definitions are not obvious to compare, at least for me!)

Is the answer to my hope `yes', and can one deduce it from what is in his book?

Answer 1

The answer is yes. Let me elaborate a bit on Jef's comment.

Let me show his claim: If $P \in E(K)$, and $\Gamma_i$ is a fibral curve of $\mathcal{E}$ intersecting the identity section, then $\tau_P(\Gamma_i) = \Gamma_i$ if and only if the section $\bar{P}$ of $\mathcal{E}$ intersects $\Gamma_i$.

The only if part: The action of $\bar{P}$ has to send the component of the identity to the component that $\bar{P}$ intersects. Note that $\bar{P}$ lies in the smooth locus.

The if part: On the other hand $\mathcal{E}^0$ has to act as the identity on the set fibral curves because images of connected sets under continuous maps are connected.

Thus, the elements of $\mathcal{E}(R)$ acting as the identity on fibral components is $\mathcal{E}^0(R)$. In order to conclude your hope we use that $\mathcal{E}(R)=E(K)$.