Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, and we suppose that $C(K) \ne \emptyset$ ($C$ may very well be defined over a proper subfield of $K$, but perhaps failing to have rational points over subfields). Then using the Abel-Jacobi map we can embed $C$ into its Jacobian $\text{Jac}(C)$, which is endowed with a Neron-Tate height $\hat{h}$. Applying this height to the image of $C$ in $\text{Jac}(C)$ we then obtain a height on $C$ itself.

Bogomolov's conjecture, proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998, asserts that there exists $\epsilon > 0$ such that the set

$$\displaystyle \{x \in C(\overline{K}) : \hat{h}(x) < \epsilon \}$$

is finite. It thus follows that there exists an algebraic point on $C$ of minimal height.

My question is this: let $S(C)$ be the set of real numbers $r$ such that $\# \{x \in C(\overline{K}) : \hat{h}(x) \leq r\} = \infty$ and let $h_{\text{min}}(C) = \inf S(C)$. Can $h_{\text{min}}(C)$ be determined by other invariants of the curve, such as the Faltings height?

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    $\begingroup$ I don't think that this is quite what you want to ask. If you use $x_0\in C(K)$ to embed $C$ into $J$, then the image of $x_0$ is the identity element of $J$, so if you vary over all $x\in C(K)$, the minimal value of $\hat h(x)$ is $0$, since you can take $x=x_0$. So maybe it would be better to define $h_\min(C)$ to the the infimum over all $\epsilon>0$ such that $$\#\{x\in C(\bar K) ; \hat h(x) <\epsilon\}=\infty.$$ $\endgroup$ – Joe Silverman Dec 29 '18 at 3:36
  • $\begingroup$ @JoeSilverman Thanks, I will make that change $\endgroup$ – Stanley Yao Xiao Dec 29 '18 at 12:12
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    $\begingroup$ The Abel-Jacobi map is defined by a choice of a degree one $\mathbb{Q}$-divisor $D_0$ defined over $K$. If you want a truly canonical choice, a standard convention is to take $D_0 = \frac{1}{2g-2} K_C = \frac{1}{2g-2}\Omega_C^1$ (indeed this was the convention in Ullmo's paper; for the other cases had been covered anyway by previous papers). Otherwise the invariant clearly depends on the choice of $D_0$, and it will be asymptotic to just the height of $D_0$ if the latter is taken to approach infinity. $\endgroup$ – Vesselin Dimitrov Dec 29 '18 at 16:55
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    $\begingroup$ Since you mention the Faltings height, yes, for a fixed $g$ it can now be proved that the latter invariant is also locked between two positive multiples (depending only on g) of the Faltings height of the curve. The easy upper bound of $12h_{\mathrm{Fal}}(C)+O_g(1)$ follows classically by the arithmetic Noether formula (and the $O_g(1)$ can be made fully explicit as well). $\endgroup$ – Vesselin Dimitrov Dec 29 '18 at 17:18
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    $\begingroup$ @VesselinDimitrov You should put that material into an answer and delete the comments. $\endgroup$ – Joe Silverman Dec 30 '18 at 2:59

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