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Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field, and let $D$ a divisor on $X$. To these data, we can associate a height function on the $\Bbb{\overline{Q}}$-rational points of $X$ $$ \operatorname{ht}_D: X(\Bbb{\overline{Q}}) \rightarrow \Bbb{R} $$ as follows. Write $D=D_1- D_2$, with $D_1,D_2$ very ample divisors on $X$, and let $\phi_{D_1},\phi_{D_2}: X \hookrightarrow \Bbb{P}^n$ be embeddings determined by $D_1,D_2$. Then $$ \operatorname{ht}_D(P):=\operatorname{ht}_{D_1}(P)-\operatorname{ht}_{D_2}(P):=\operatorname{h}(\phi_{D_1}(P))-\operatorname{h}(\phi_{D_2}(P)) $$ where $\operatorname{h}: \Bbb{P}^n \rightarrow \Bbb{R}$ is the usual height function on projective space. One can check that, up to a bounded function, $\operatorname{ht}_D$ does not depend on the choices of $D_1,D_2$.

Now I know that if $\deg(D) >0$, the height function $\operatorname{ht}_D$ is bounded from below, and if $\deg(D) < 0$, then $\operatorname{ht}_D$ is bounded from above.

$\textbf{Question:}$ What happens if $\deg(D)=0$? Is there a curve $X$ as above and a divisor $D$, with $\deg(D)=0$ such that $\operatorname{ht}_D$ is not bounded from below/above?

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The question as to whether $\operatorname{ht}_D$ for a degree zero divisor $D$ is bounded depends on the class of $D$ in the Picard group of $X$.

Theorem: Let $X$ be a smooth, projective curve over a number field, and $D$ a divisor on $X$. Then $\operatorname{ht}_D$ is bounded, if and only if $D$ is a torsion element of the Picard group $\operatorname{Pic}(X)$ of $X$.

More generally, the theorem is true for an arbitrary smooth, projective variety.

The "if" part is clear, and the only if part is proved in

J.-P. Serre, Lectures on the Mordell-Weil Theorem, Theorem 3.11.

The most important fact for the proof seems to be that the Jacobian of $X$ is principally polarised, i.e. it is isomorphic to its dual abelian variety.

The above theorem is somewhat interesting in my opinion, partly because it resembles the well-known result that on an abelian variety $A$ a point is a torsion point, if and only if its Néron-Tate height (associated to an ample, symmetric divisor) vanishes.

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