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 $X$ the Néron-Tate canonical height $\hat{h}_A(x,\mathcal{L})$ equal to $\mathrm{deg}_X((x,\mathcal{L})^*\mathcal{P}_A$ with the Poincaré bundle $\mathcal{P}_A$?

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 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.?

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?

Edit: Another question: Why is for a curve $X$ the Néron-Tate canonical height $\hat{h}_A(x,\mathcal{L})$ equal to $\mathrm{deg}_X((x,\mathcal{L})^*\mathcal{P}_A$ with the Poincaré bundle $\mathcal{P}_A$?

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 $X$ the Néron-Tate canonical height $\hat{h}_A(x,\mathcal{L})$ equal to $\mathrm{deg}_X((x,\mathcal{L})^*\mathcal{P}_A$ with the Poincaré bundle $\mathcal{P}_A$?

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.?

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?

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?

Edit: Another question: Why is for a curve $X$ the Néron-Tate canonical height $\hat{h}_A(x,\mathcal{L})$ equal to $\mathrm{deg}_X((x,\mathcal{L})^*\mathcal{P}_A$ with the Poincaré bundle $\mathcal{P}_A$?