## 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.?

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 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? – Ariyan Javanpeykar Jul 18 at 20:26 I clarified question 2. Thank you for your response. – Timo Keller Jul 20 at 6:33

See Chapter 2 of the book by E. Bombieri and W. Gubler, Heights in Diophantine Geometry. They begin classically: they first define local heights (§2.2), then global heights (§2.3), and finally compare their global heights with Weil's definition (§2.4). Later, §2.7 gives the alternate point of view of Arakelov geometry on local heights, which should contain what you want. Indeed, they define a local height with respect to a Néron divisor (2.7.9) while Example 2.7.20 explains how models over DVRs give rise to Néron divisors.

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Lang, Fundamentals of Diophantine Geometry, Chapter 3, Section 3 (Heights in Function Fields) does global heights as intersection indices, which I realize isn't quite what you want, but probably the ideas here can be used to prove the local case and get a local height.

Lang does consider local heights in Chapter 11, Section 5 (Neron Functions as Intersection Multiplicities), but only on abelian varieties. However, he proves the much stronger result that not only does the intersection index give a local Weil height, but it transforms functorially relative to the group law, hence is a Neron (canonical) local height.

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