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(transcendence of canonical heights)

Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always lie in $\bar{\mathbf{Q}}$?

Edit: I am looking for a reference that the height is always rational.

Edit 2: Perhaps the following is true: Let $\mathcal{A}/X$ be the Néron model of $A/K$. Then $\hat{h}(x,\mathcal{L}) = \mathrm{deg}((x,\mathcal{L})^*\mathcal{P}_\mathcal{A})$ for the Poincare bundle $\mathcal{P}_\mathcal{A}$ of $\mathcal{A}$, $x \in \mathcal{A}(X)$, $\mathcal{L} \in \mathcal{A}^\vee(X)$ and $\mathrm{deg}: \mathrm{Pic}(X) \to \mathbf{Z}$.

(transcendence of canonical heights)

Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always lie in $\bar{\mathbf{Q}}$?

Edit: I am looking for a reference that the height is always rational.

Edit 2: Perhaps the following is true: Let $\mathcal{A}/X$ be the Néron model of $A/K$. Then $\hat{h}(x,\mathcal{L}) = \mathrm{deg}((x,\mathcal{L})^*\mathcal{P}_\mathcal{A})$ for the Poincare bundle $\mathcal{P}_\mathcal{A}$ of $\mathcal{A}$, $x \in \mathcal{A}(X)$, $\mathcal{L} \in \mathcal{A}^\vee(X)$ and $\mathrm{deg}: \mathrm{Pic}(X) \to \mathbf{Z}$.

(transcendence of canonical heights)

Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always lie in $\bar{\mathbf{Q}}$?

Edit: I am looking for a reference that the height is always rational.

Edit 2: Perhaps the following is true: Let $\mathcal{A}/X$ be the Néron model of $A/K$. Then $\hat{h}(x,\mathcal{L}) = \mathrm{deg}((x,\mathcal{L})^*\mathcal{P}_\mathcal{A})$ for the Poincare bundle $\mathcal{P}_\mathcal{A}$ of $\mathcal{A}$, $x \in \mathcal{A}$, $\mathcal{L} \in \mathcal{A}^\vee$ and $\mathrm{deg}: \mathrm{Pic}(X) \to \mathbf{Z}$.

(transcendence of canonical heights)

Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always lie in $\bar{\mathbf{Q}}$?

Edit: I am looking for a reference that the height is always rational.

Edit 2: Perhaps the following is true: Let $\mathcal{A}/X$ be the Néron model of $A/K$. Then $\hat{h}(x,\mathcal{L}) = \mathrm{deg}((x,\mathcal{L})^*\mathcal{P}_\mathcal{A})$ for the Poincare bundle $\mathcal{P}_\mathcal{A}$ of $\mathcal{A}$, $x \in \mathcal{A}(X)$, $\mathcal{L} \in \mathcal{A}^\vee(X)$ and $\mathrm{deg}: \mathrm{Pic}(X) \to \mathbf{Z}$.

(transcendence of canonical heights)

Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always lie in $\bar{\mathbf{Q}}$?

Edit: I am looking for a reference that the height is always rational.

(transcendence of canonical heights)

Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always lie in $\bar{\mathbf{Q}}$?

Edit: I am looking for a reference that the height is always rational.

Edit 2: Perhaps the following is true: Let $\mathcal{A}/X$ be the Néron model of $A/K$. Then $\hat{h}(x,\mathcal{L}) = \mathrm{deg}((x,\mathcal{L})^*\mathcal{P}_\mathcal{A})$ for the Poincare bundle $\mathcal{P}_\mathcal{A}$ of $\mathcal{A}$, $x \in \mathcal{A}$, $\mathcal{L} \in \mathcal{A}^\vee$ and $\mathrm{deg}: \mathrm{Pic}(X) \to \mathbf{Z}$.