# Points of low height and low degree

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?

• 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? – Ariyan Javanpeykar Jan 28 at 9:11
• @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? – Stanley Yao Xiao Jan 28 at 11:38