Are there known examples of rational points on elliptic curves/abelian varieties over number fields with transcendental canonical height? Thanks.
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$\begingroup$ Why do you want to know? E.g., why is this any different than asking whether any old real number defined as a limit is transcendental? $\endgroup$– Pete L. ClarkCommented Feb 22, 2011 at 22:49
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$\begingroup$ Good point! My specific interest is: Is the canonical height a period or not(say as defined in Kontsevich-Zagier)? $\endgroup$– SGPCommented Feb 22, 2011 at 23:29
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4$\begingroup$ If the analytic rank of an elliptic curve over $\mathbf{Q}$ is 1 then the height of the Heegner point is a period by Gross-Zagier, I believe. But I don't think anything like this is known for higher rank. But that is completely different from your question... $\endgroup$– Chris WuthrichCommented Feb 23, 2011 at 1:25
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$\begingroup$ Could you elaborate on the Gross-Zagier comment? why is the height of a Heegner point a period? Thanks! $\endgroup$– SGPCommented Feb 24, 2011 at 23:43
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2 Answers
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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.
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$\begingroup$ Many thanks for the commentsl It is very interesting to learn of the results in algebraic dynamics! $\endgroup$– SGPCommented Apr 16, 2011 at 11:24
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$\begingroup$ "For Abelian varieties, it follows from the explicit formulae for the Néron local pairing that the canonical height is even rational." Can you give me a precise reference for this claim? $\endgroup$– user19475Commented Dec 4, 2013 at 17:55
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1$\begingroup$ Néron's approach to the Néron-Tate height expresses it as a sum of local terms; almost all of them are zero, and (over function fields) all of them are rational numbers, whose denominators are controlled by the group of connected components of the Néron model. (Of course, this does not hold over number fields, because of the archimedean contribution.) $\endgroup$– ACLCommented Dec 5, 2013 at 1:23
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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!)
answered Dec 5, 2013 at 0:47