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In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell conjecture, and in higher dimensions, the Lang conjecture states that if $X(\mathbb{C})^{an}$ is hyperbolic, then $|X(k)|$ is finite. I would be interested to generalize this train of thought to the category of schemes up to $\mathbb{A}^1$-homotopy. Concretely, my question is whether or not there exists a (conjectural) geometrical condition on $[X]$, an $\mathbb{A}^1$-homotopy class, such that the set of morphism $\text{Hom}_{\mathbb{A}^1\text{-hom}}([\text{Spec } k],[X])$ in the category of schemes up to $\mathbb{A}^1$-homotopy is finite.

A naive translation of Faltings Theorem of course doesn't work, as for instance $\mathbb{A}^1_{\mathbb{Q}}$ has infintely many rational points, but $\text{Spec }\mathbb{Q}$ does not.

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    $\begingroup$ This mapping space is (I think) the same as the space of rationally connected components. In particular naive Mordell still works, since no two different points of a curve of genus $\ge 1$ are rationally connected! $\endgroup$ – Dmitry Vaintrob Feb 21 at 3:46

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