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Let $A$ be an abelian variety over some number field $K$. We know the $\mathbb{C}$-points of $A$ form a complex analytic manifold, so $A(\mathbb{C})$ is a smooth manifold in fact. It then makes sense to talk about PDEs on such a space (ignoring local-global issues at the moment). Now, we also have our canonical height function on $A$. My question is whether there is any connection between our height function and these PDEs we can create. Obviously, the height function is very not-continuous, so I am more wondering if anyone has modified the height function in someway to make it "work" with some specific PDE.

Goal: I am interested to see if such a thing could be used to prove facts about small points on abelian varieties. Ideally, there would be some analog of the heat equation with the "temperature distribution function" tracking small points nicely is my idea.

Thank you for any references or pointing out the impossibility of such things!

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    $\begingroup$ Welcome new contributor. What do you mean by PDEs on the Abelian variety? Do you have in mind some particular integrable system? There is an analytic aspect of heights that forms part of the subject of “Arakelov geometry.” That may be a starting point for what you describe. $\endgroup$ Commented Oct 31 at 20:36
  • $\begingroup$ Hi! Thank you for the quick response. In short, I am mostly wondering if the canonical height function can be modified to be continuous/smooth/$L^2$ (when considered as a function on $A(\mathbb{C})$. I have looked into Gillet and Soulé's height construction, but I haven't found anything saying such (Obviously, I could easily be missing something right in front of me as well). $\endgroup$ Commented Oct 31 at 20:47

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