# Finiteness results in the category of schemes up to $\mathbb{A}^1$-homotopy

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.

• 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! – Dmitry Vaintrob Feb 21 at 3:46