# questions on Néron-Tate canonical height

I have three questions regarding height pairings:

1. In [Serre, Lectures on the Mordell-Weil theorem], p. 85 f., it is stated that the following function is a local height function: "Let $V/R$ be a scheme over a discrete valuation ring, $D$ a divisor on $V$ and $x \in V(R)$. Assume $x \not\in \mathrm{supp}(D)$. Define $m(x,D)$ as follows: a) If $x$ does not intersect $D$, $m(x,D) = 0$. b) If $x$ intersects $D$ in a point $x_0$ of the special fibre, then $m(x,D)$ is the intersection multiplicity of $x$ and $D$ and $x_0$. Then $x \mapsto m(x,D)$ is a local height associated to $D$." Does anyone have a reference for this?

2. Why is for a curve $X/k$ and an Abelian variety $B/k$ the Néron-Tate canonical height of the constant Abelian variety $B \times_k X$ over $X$ and $x: X \to B$ and $\mathcal{L}: X \to B^\vee$ $$\hat{h}(x,\mathcal{L}) = \mathrm{deg}_X((x,\mathcal{L})^*\mathcal{P}_B)$$ with the Poincaré bundle $\mathcal{P}_B \in \mathrm{Pic}(B \times_k B^\vee)$? The degree function $\mathrm{deg}: \mathrm{Pic}(X) \to \mathbf{Z}$ is the usual one for a curve.

3. We have (functoriality of the height) $\hat{h}_{f^*(\mathcal{M})} = \hat{h}_{\mathcal{M}} \circ f$. Can one prove this also with the other definition of the height in 2.?

-
I hope you don't mind me asking, but could you elaborate a bit on Question 2? Is $A$ the Jacobian of $X$? Is $\mathcal L$ a line bundle on $A$? Is $x$ a point on $X$? And what is $\deg ((x,\mathcal L)^\ast \mathcal P_A$? I don't understand the notation $(x,\mathcal L)^\ast$ here. Are you taking the degree of a line bundle on $X$ in the usual sense? –  Ari Jul 18 '12 at 20:26
I clarified question 2. Thank you for your response. –  Timo Keller Jul 20 '12 at 6:33