transcendence of canonical heights Are there known examples of rational points on elliptic curves/abelian varieties
over number fields with transcendental canonical height? Thanks.
 A: This is a comment, rather than an answer.
Z. Chadzidakis and E. Hrushovski discuss this question in Section 4.19 of their paper
Difference fields and descent in algebraic dynamics - I,
They mention a suggestion of Silverman that for Abelian varieties over number fields,
the canonical Néron-Tate height might be transcendental; I do not of a precise
reference for this suggestion. 
I would presume that known examples are rare.
However, the paper of Chadzidakis and Hrushovski is about canonical heights
in algebraic dynamics over function fields.
Then, the picture is quite different for they prove that 
the canonical height is ''often'' algebraic.
For Abelian varieties, it follows from the explicit formulae for the Néron local pairing
that the canonical height is even rational.
A: The answer to the question is "no, there are currently no known examples." 
On the other hand, I think it's reasonable to conjecture that over a number field, the canonical height of a non-torsion point on an elliptic curve is transcendental. I've mentioned this in conversation to various people over the years, but I don't know that it appears in print (other than in that comment in the Chadzidakis-Hrushovski paper). And with appropriate non-degeneracy conditions, it should also be true on abelian varieties.
In response to Pete's question of why one would want to know such a result for canonical heights, as opposed to any miscellaneous number defined as a limit, I would say it is because these canonical heights appear in the Birch-Swinnerton-Dyer formula, so are (conjecturally) related to values of $L$-series. So if they were algebraic, it would mean that (say) $L'(E,1)/\Omega$ is algebraic for curves of rank 1. This would be quite interesting.
Regarding the question of canonical heights in dynamics, it's easy to produce examples where the (multiplicative) canonical height is an integer. A reasonable guess is the following: For a rational map $f\in\overline{\mathbb{Q}}(x)$ of degree at least 2 and a point $\alpha\in\mathbb{P}^1(\overline{\mathbb{Q}})$ with infinite forward orbit that lies in the Julia set of $f$ in $\mathbb{P}^1(\mathbb{C})$, both the logarithmic  and the multiplicative  canonical heights $\hat h_f(\alpha)$ and $\exp(\hat h_f(\alpha))$ are transcendental. This would cover the elliptic curve case, taking $f$ to be a Lattes map and using the fact that the Julia set of a Lattes map is all of $\mathbb{P}^1(\mathbb{C})$.
Finally, I want to mention that there is an old result of Daniel Bertrand in which he proves that the $p$-adic canonical height of a non-torsion point on a CM elliptic curve is transcendental. (This is how he proved that it is nonzero!)
