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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 $\mathrm{deg}: \mathrm{Pic}(X) \to \mathbf{Z}$.

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    $\begingroup$ It lies in $\mathbb{Q}$, basically by intersection theory on the Neron model. This is mentioned in ACL's answer to the question you link to. For elliptic curves, this is contained in Theorem 9.3 (due to Manin) in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves. $\endgroup$ – Vesselin Dimitrov Dec 4 '13 at 17:35
  • $\begingroup$ Thanks. I have asked a question in the mentioned linked question/answer. $\endgroup$ – TKe Dec 4 '13 at 18:14
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    $\begingroup$ The canonical height is a sum of local heights. At the non-archimedean places, the local contribution is $r\log q$, where $r\in\mathbb{Q}$ and $q$ is the norm of the prime. So over a function field, the logarithmic canonical height is $r\log(n)$, and the exponential height is algebraic. For number fields, the contribution from the archimedean places are logs of absolute values of theta functions (more or less). For a further discussion of the number field case, see the answer that I just posted at mathoverflow.net/questions/56331/… $\endgroup$ – Joe Silverman Dec 5 '13 at 0:53
  • $\begingroup$ Re. edit 2: you must be careful as the Poincare bundle does not in general extend over the whole Neron model, but only over the identity component. However, as long as you multiply your section by a suitable positive integer to get in the identity component, then I think the proposed equality holds. It should be in Moret-Bailly’s article in Asterisque 127 in the number field case; IIRC same proof works (more easily) in function field case. $\endgroup$ – David Holmes Sep 30 '14 at 19:22

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