[Math] questions on Néron-Tate canonical height

arithmetic-geometryheightsnt.number-theoryreference-request

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.?

Best Answer

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.

Related Question