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Let $K$ be a number field, and let $C$ be an algebraic curve of genus $g \geq 2$ defined over $K$. We define $d_C$ to be the positive integer with the following property: there exists at most finitely many $\overline{\mathbb{Q}}$-points of $x \in C$ with the property that $[K(x) : K] \leq d_C$, and for any integer $d > d_C$, there exists infinitely many points $x \in C(\overline{\mathbb{Q}})$ such that $[K(x) : K] \leq d$. Note that this is well defined by Faltings' theorem. Here the notation $K(x)$ refers to adjoining all coordinates of $x$ to $K$. We say that $x \in C(\overline{\mathbb{Q}})$ is a low degree point of $C$ if $[K(x) : K] \leq d_C$. Denote by $S_C$ the set of low degree points.

We now consider the stable canonical height $h$ on $C(\overline{\mathbb{Q}})$ given by Arakelov theory and the canonical sheaf. By Bogomolov's conjecture (i.e., theorem of Shou-Wu Zhang and Emmanuel Ullmo), there exists a number $\varepsilon_C > 0$ such that the set $\{x \in C(\overline{\mathbb{Q}}) : h(x) < \varepsilon_C\}$ is finite, and for all $\delta > \varepsilon_C$, the set $\{x \in C(\overline{\mathbb{Q}}) : h(x) < \delta\}$ is Zariski-dense. Call the elements in $T_C = \{x \in C(\overline{\mathbb{Q}}) : h(x) < \varepsilon_C\}$ the set of low height points.

My question concerns the intersection of $T_C \cap S_C$, or points which are both low degree and low height. Clearly, this intersection can be empty: for example, when $C$ is a hyperelliptic curve we have $d_C = 1$, so $S_C = C(K)$, which can already be empty. Now suppose that $C(K) \ne \emptyset$. Do we still expect $S_C \cap T_C = \emptyset$ typically?

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  • $\begingroup$ Let $C$ be a curve over $K$. Choose $L/K$ a finite field extension such that $T_C$ is non-empty and $C(L)$ contains a point $P$ from $T_C$. Now, $S_{C_L}$ contains $C(L)$ and thus $P$. so the intersection $S_{C_L}\cap T_{C}$ is non-empty. Or did I misunderstand something? $\endgroup$ – Ariyan Javanpeykar Jan 28 at 9:11
  • $\begingroup$ @AriyanJavanpeykar Hi Ariyan: you can certainly give many examples using your construction, by simply redefining the field of definition. However my question is about typical behaviour: that is, among all curves of a given genus defined over a field $K$ with at least one $K$-point (this is already a fairly small subset of all curves, probably), do we expect there to be a point of both low height and low degree typically? $\endgroup$ – Stanley Yao Xiao Jan 28 at 11:38

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